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from manimlib import *
from manimlib.mobject.svg.old_tex_mobject import *
from custom.backdrops import *
from custom.banner import *
from custom.characters.pi_creature import *
from custom.characters.pi_creature_animations import *
from custom.characters.pi_creature_scene import *
from custom.deprecated import *
from custom.drawings import *
from custom.end_screen import *
from custom.filler import *
from custom.logo import *
from custom.opening_quote import *
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This project contains the code used to generate the explanatory math videos found on [3Blue1Brown](https://www.3blue1brown.com/).
This almost entirely consists of scenes generated using the library [Manim](https://github.com/3b1b/manim). See also the community maintained version at [ManimCommunity](https://github.com/ManimCommunity/manim/).
Note, while the library Manim itself is open source and under the MIT license, the contents of this project are intended only to be used for 3Blue1Brown videos themselves.
Copyright © 2022 3Blue1Brown
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directories:
mirror_module_path: True
removed_mirror_prefix: "/Users/grant/cs/videos/"
output: "/Users/grant/3Blue1Brown Dropbox/3Blue1Brown/videos"
raster_images: "/Users/grant/3Blue1Brown Dropbox/3Blue1Brown/images/raster"
vector_images: "/Users/grant/3Blue1Brown Dropbox/3Blue1Brown/images/vector"
pi_creature_images: "/Users/grant/3Blue1Brown Dropbox/3Blue1Brown/images/pi_creature/svg"
sounds: "/Users/grant/3Blue1Brown Dropbox/3Blue1Brown/sounds"
data: "/Users/grant/3Blue1Brown Dropbox/3Blue1Brown/data"
temporary_storage: "/Users/grant/3Blue1Brown Dropbox/3Blue1Brown/manim_cache"
universal_import_line: "from manim_imports_ext import *"
# tex:
# executable: "xelatex -no-pdf"
# template_file: "ctex_template.tex"
# intermediate_filetype: "xdv"
style:
font: "CMU Serif"
text_alignment: "CENTER"
background_color: "#000000"
camera_resolutions:
default_resolution: "4k"
window_position: UR
window_monitor: 0
full_screen: False
embed_exception_mode: "Minimal"
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#!/usr/bin/env python
import inspect
import os
import sys
import importlib
from manimlib.config import get_module
from manimlib.extract_scene import is_child_scene
def get_sorted_scene_classes(module_name):
module = get_module(module_name)
if hasattr(module, "SCENES_IN_ORDER"):
return module.SCENES_IN_ORDER
# Otherwise, deduce from the order in which
# they're defined in a file
importlib.import_module(module.__name__)
line_to_scene = {}
name_scene_list = inspect.getmembers(
module,
lambda obj: is_child_scene(obj, module)
)
for name, scene_class in name_scene_list:
if inspect.getmodule(scene_class).__name__ != module.__name__:
continue
lines, line_no = inspect.getsourcelines(scene_class)
line_to_scene[line_no] = scene_class
return [
line_to_scene[index]
for index in sorted(line_to_scene.keys())
]
def stage_scenes(module_name):
scene_classes = get_sorted_scene_classes(module_name)
if len(scene_classes) == 0:
print("There are no rendered animations from this module")
return
# TODO, fix this
animation_dir = os.path.join(
os.path.expanduser('~'),
"Dropbox/3Blue1Brown/videos/2021/holomorphic_dynamics/videos"
)
#
files = os.listdir(animation_dir)
sorted_files = []
for scene_class in scene_classes:
scene_name = scene_class.__name__
clips = [f for f in files if f.startswith(scene_name + ".")]
for clip in clips:
sorted_files.append(os.path.join(animation_dir, clip))
# Partial movie file directory
# movie_dir = get_movie_output_directory(
# scene_class, **output_directory_kwargs
# )
# if os.path.exists(movie_dir):
# for extension in [".mov", ".mp4"]:
# int_files = get_sorted_integer_files(
# pmf_dir, extension=extension
# )
# for file in int_files:
# sorted_files.append(os.path.join(pmf_dir, file))
# else:
# animation_subdir = os.path.dirname(animation_dir)
count = 0
while True:
staged_scenes_dir = os.path.join(
animation_dir,
os.pardir,
"staged_scenes_{}".format(count)
)
if not os.path.exists(staged_scenes_dir):
os.makedirs(staged_scenes_dir)
break
# Otherwise, keep trying new names until
# there is a free one
count += 1
for count, f in reversed(list(enumerate(sorted_files))):
# Going in reversed order means that when finder
# sorts by date modified, it shows up in the
# correct order
symlink_name = os.path.join(
staged_scenes_dir,
"Scene_{:03}_{}".format(
count, f.split(os.sep)[-1]
)
)
os.symlink(f, symlink_name)
if __name__ == "__main__":
if len(sys.argv) < 2:
raise Exception("No module given.")
module_name = sys.argv[1]
stage_scenes(module_name)
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from manim_imports_ext import *
from _2022.wordle.simulations import *
# Scene types
class WordleScene(Scene):
n_letters = 5
grid_height = 6
font_to_square_height = 65
grid_center = ORIGIN
secret_word = None
color_map = {
0: "#797C7E", # GREY
1: "#C6B566", # YELLOW
2: GREEN_D, # GREEN
}
uniform_prior = False
wordle_based_prior = False
freq_prior = True
reveal_run_time = 2
reveal_lag_ratio = 0.5
CONFIG = {"random_seed": None}
def setup(self):
self.all_words = self.get_word_list()
self.priors = self.get_priors()
if self.secret_word is None:
s_words = get_word_list(short=True)
self.secret_word = random.choice(s_words)
self.guesses = []
self.patterns = []
self.possibilities = self.get_initial_possibilities()
self.add_grid()
def get_word_list(self):
return get_word_list()
def get_initial_possibilities(self):
return get_word_list()
def get_priors(self):
words = self.all_words
if self.uniform_prior:
return dict((w, 1) for w in words)
elif self.wordle_based_prior:
return get_true_wordle_prior()
else:
return get_frequency_based_priors()
def get_pattern(self, guess):
return get_pattern(guess, self.secret_word)
def get_current_entropy(self):
weights = get_weights(self.possibilities, self.priors)
return entropy_of_distributions(weights)
##
def add_grid(self):
buff = 0.1
row = Square(side_length=1).get_grid(1, self.n_letters, buff=buff)
grid = row.get_grid(6, 1, buff=buff)
grid.set_height(self.grid_height)
grid.move_to(self.grid_center)
grid.set_stroke(WHITE, 2)
grid.words = VGroup()
grid.pending_word = VGroup()
grid.add(grid.words, grid.pending_word)
grid.pending_pattern = None
grid.add_updater(lambda m: m)
self.grid = grid
self.add(grid)
def get_curr_row(self):
return self.grid[len(self.grid.words)]
def get_curr_square(self):
row = self.get_curr_row()
return row[len(self.grid.pending_word)]
def add_letter(self, letter):
grid = self.grid
if len(grid.pending_word) == len(grid[0]):
return
letter_mob = self.get_letter_in_square(letter, self.get_curr_square())
grid.pending_word.add(letter_mob)
def get_letter_in_square(self, letter, square):
font_size = self.font_to_square_height * square.get_height()
letter_mob = Text(letter.upper(), font="Consolas", font_size=font_size)
letter_mob.move_to(square)
return letter_mob
def delete_letter(self):
if len(self.grid.pending_word) == 0:
return
letter_mob = self.grid.pending_word[-1]
self.grid.pending_word.remove(letter_mob)
def add_word(self, word, wait_time_per_letter=0.1):
for letter in word:
self.add_letter(letter)
self.wait(
wait_time_per_letter,
ignore_presenter_mode=True
)
def pending_word_as_string(self):
return "".join(
t.text.lower()
for t in self.grid.pending_word
)
def is_valid_guess(self):
guess = self.pending_word_as_string()
return guess in self.all_words
def reveal_pattern(self, pattern=None, animate=True):
grid = self.grid
guess = self.pending_word_as_string()
if not self.is_valid_guess():
self.shake_word_out()
return False
if pattern is None:
pattern = self.get_pattern(guess)
if pattern is None:
return False
self.show_pattern(pattern, animate=animate)
self.guesses.append(guess)
self.patterns.append(pattern)
grid.words.add(grid.pending_word.copy())
grid.pending_word.set_submobjects([])
grid.pending_pattern = None
self.refresh_possibilities(guess, pattern)
# Win condition
if self.has_won():
self.win_animation()
return True
def refresh_possibilities(self, guess, pattern):
self.possibilities = get_possible_words(
guess, pattern, self.possibilities
)
def shake_word_out(self):
row = self.get_curr_row()
c = row.get_center().copy()
func = bezier([0, 0, 1, 1, -1, -1, 0, 0])
self.play(UpdateFromAlphaFunc(
VGroup(row, self.grid.pending_word),
lambda m, a: m.move_to(c + func(a) * RIGHT),
run_time=0.5,
))
self.grid.pending_word.set_submobjects([])
def show_pattern(self, pattern, animate=False, added_anims=[]):
row = self.get_curr_row()
colors = self.get_colors(pattern)
if animate:
self.animate_color_change(row, self.grid.pending_word, colors, added_anims)
else:
self.set_row_colors(row, colors)
self.grid.pending_pattern = pattern
def set_row_colors(self, row, colors):
for square, color in zip(row, colors):
square.set_fill(color, 1)
def animate_color_change(self, row, word, colors, added_anims=[]):
colors.extend((len(row) - len(colors)) * [self.color_map[0]])
for square, color in zip(row, colors):
square.future_color = color
def alpha_func(mob, alpha):
if not hasattr(mob, 'initial_height'):
mob.initial_height = mob.get_height()
mob.set_height(
mob.initial_height * max(abs(interpolate(1, -1, alpha)), 1e-6),
stretch=True
)
if isinstance(mob, Square) and alpha > 0.5:
mob.set_fill(mob.future_color, 1)
mobjects = self.mobjects
self.play(
*(
LaggedStart(
*(
UpdateFromAlphaFunc(sm, alpha_func)
for sm in mob
),
lag_ratio=self.reveal_lag_ratio,
run_time=self.reveal_run_time,
)
for mob in (row, word)
),
*added_anims
)
self.clear()
self.add(*mobjects)
def get_colors(self, pattern):
return [self.color_map[key] for key in pattern_to_int_list(pattern)]
def win_animation(self):
grid = self.grid
row = grid[len(grid.words) - 1]
letters = grid.words[-1]
mover = VGroup(*(
VGroup(square, letter)
for square, letter in zip(row, letters)
))
y = row.get_y()
bf = bezier([0, 0, 1, 1, -1, -1, 0, 0])
self.play(
LaggedStart(*(
UpdateFromAlphaFunc(sm, lambda m, a: m.set_y(y + 0.2 * bf(a)))
for sm in mover
), lag_ratio=0.1, run_time=1.5),
LaggedStart(*(
Flash(letter, line_length=0.1, flash_radius=0.4)
for letter in letters
), lag_ratio=0.3, run_time=1.5),
)
def has_won(self):
return len(self.patterns) > 0 and self.patterns[-1] == 3**5 - 1
@staticmethod
def get_grid_of_words(all_words, n_rows, n_cols, dots_index=-5, sort_key=None, font_size=24):
if sort_key:
all_words = list(sorted(all_words, key=sort_key))
subset = all_words[:n_rows * n_cols]
show_ellipsis = len(subset) < len(all_words)
if show_ellipsis:
subset[dots_index] = "..." if n_cols == 1 else "....."
subset[dots_index + 1:] = all_words[dots_index + 1:]
full_string = ""
for i, word in zip(it.count(1), subset):
full_string += str(word)
if i % n_cols == 0:
full_string += " \n"
else:
full_string += " "
full_text_mob = Text(full_string, font="Consolas", font_size=font_size)
result = VGroup()
for word in subset:
part = full_text_mob.get_part_by_text(word)
part.text = word
result.add(part)
if show_ellipsis and n_cols == 1:
result[dots_index].rotate(PI / 2)
result[dots_index].next_to(result[dots_index - 1], DOWN, SMALL_BUFF)
result[dots_index + 1:].next_to(result[dots_index], DOWN, SMALL_BUFF)
result.set_color(GREY_A)
result.words = subset
return result
@staticmethod
def patterns_to_squares(patterns, color_map=None):
if color_map is None:
color_map = WordleScene.color_map
row = Square().get_grid(1, 5, buff=SMALL_BUFF)
rows = row.get_grid(len(patterns), 1, buff=SMALL_BUFF)
rows.set_stroke(WHITE, 1)
for pattern, row in zip(patterns, rows):
for square, key in zip(row, pattern_to_int_list(pattern)):
square.set_fill(color_map[key], 1)
return rows
# Interactive parts
def on_key_press(self, symbol, modifiers):
try:
char = chr(symbol)
except OverflowError:
log.warning("The value of the pressed key is too large.")
return
is_letter = (ord('a') <= ord(char) <= ord('z'))
if is_letter:
self.add_letter(char)
elif symbol == 65288: # Delete
self.delete_letter()
elif symbol == 65293: # Enter
self.reveal_pattern()
if char == 'q' and modifiers == 1:
self.delete_letter()
self.quit_interaction = True
if not is_letter:
super().on_key_press(symbol, modifiers)
class WordleSceneWithAnalysis(WordleScene):
grid_center = [-1.75, 1, 0]
grid_height = 4.5
look_two_ahead = False
show_prior = True
n_top_picks = 13
entropy_color = TEAL_C
prior_color = BLUE_C
weight_to_prob = 3.0
pre_computed_first_guesses = []
def setup(self):
self.wait_to_proceed = True
super().setup()
self.show_possible_words()
self.add_count_title()
self.add_guess_value_grid_title()
self.init_guess_value_grid()
self.info_labels = VGroup()
self.add(self.info_labels)
def construct(self):
self.show_guess_values()
def add_guess_value_grid_title(self):
titles = VGroup(
Text("Top picks"),
Text("E[Info.]", color=self.entropy_color),
)
if self.look_two_ahead:
titles.add(OldTexText("E[Info$_2$]", color=self.entropy_color)[0])
if self.show_prior:
titles.add(OldTex("p(\\text{word})", color=self.prior_color))
titles.scale(0.7)
titles.arrange(RIGHT, buff=MED_LARGE_BUFF)
titles.set_max_width(5)
low_y = titles[0][0].get_bottom()[1]
for title in titles:
first = title.family_members_with_points()[0]
title.shift((low_y - first.get_bottom()[1]) * UP)
underline = Underline(title)
underline.match_y(first.get_bottom() + 0.025 * DOWN)
underline.set_stroke(WHITE, 2)
underline.scale(1.1)
title.add_to_back(underline)
title.set_backstroke()
underline.set_stroke(GREY_C, 2)
titles.to_edge(UP, buff=MED_SMALL_BUFF)
titles.to_edge(RIGHT, buff=MED_SMALL_BUFF)
self.add(titles)
self.guess_value_grid_titles = titles
def add_count_title(self):
title = VGroup(
Text("# Possibilities"),
Text("/"),
Text("Uncertainty", color=self.entropy_color),
)
title.arrange(RIGHT, buff=SMALL_BUFF)
title.match_width(self.count_label).scale(1.1)
title.next_to(self.count_label, UP, buff=MED_LARGE_BUFF)
self.count_title = title
self.add(title)
def init_guess_value_grid(self):
titles = self.guess_value_grid_titles
line = Line().match_width(titles)
line.set_stroke(GREY_C, 1)
lines = line.get_grid(self.n_top_picks, 1, buff=0.5)
lines.next_to(titles, DOWN, buff=0.75)
self.guess_value_grid_lines = lines
self.guess_value_grid = VGroup()
def get_count_label(self):
score = len(self.grid.words)
label = VGroup(
Integer(len(self.possibilities), edge_to_fix=UR),
Text("Pos,"),
DecimalNumber(self.get_current_entropy(), edge_to_fix=UR, color=self.entropy_color),
Text("Bits", color=self.entropy_color),
)
label.arrange(
RIGHT,
buff=MED_SMALL_BUFF,
aligned_edge=UP,
)
label.scale(0.6)
label.next_to(self.grid[score], LEFT)
return label
def reveal_pattern(self, pattern=None, animate=True):
is_valid_guess = self.is_valid_guess()
if is_valid_guess:
self.isolate_guessed_row()
did_fill = super().reveal_pattern(pattern, animate)
if not did_fill:
return False
if self.presenter_mode:
while self.wait_to_proceed:
self.update_frame(1 / self.camera.frame_rate)
self.wait_to_proceed = True
if is_valid_guess and not self.has_won():
self.show_possible_words()
self.wait()
self.show_guess_values()
if self.has_won():
self.play(
FadeOut(self.guess_value_grid, RIGHT),
FadeOut(self.guess_value_grid_titles, RIGHT),
)
def show_pattern(self, pattern, *args, **kwargs):
guess = self.pending_word_as_string()
new_possibilities = get_possible_words(
guess, pattern, self.possibilities
)
for word_mob, word, bar in zip(self.shown_words, self.shown_words.words, self.prob_bars):
if word not in new_possibilities and word != "...":
word_mob.set_fill(RED, 0.5)
bar.set_opacity(0.2)
self.show_pattern_information(guess, pattern, new_possibilities)
super().show_pattern(pattern, *args, **kwargs)
def show_pattern_information(self, guess, pattern, new_possibilities):
# Put bits label next to pattern
weights = get_weights(self.possibilities, self.priors)
prob = sum(
weight for word, weight in zip(self.possibilities, weights)
if word in new_possibilities
)
info = -math.log2(prob)
ref = self.count_label[2:]
info_label = VGroup(
DecimalNumber(info),
Text("Bits")
)
info_label.set_color(RED)
info_label.arrange(RIGHT)
info_label.match_height(ref)
info_label.next_to(self.get_curr_row(), RIGHT, buff=MED_SMALL_BUFF)
info_label.match_y(ref)
self.info_labels.add(info_label)
def isolate_guessed_row(self):
guess = self.pending_word_as_string()
rows = self.guess_value_grid
row_words = [row[0].text for row in rows]
if guess in row_words:
row = rows[row_words.index(guess)]
rows.set_opacity(0.2)
row[:-1].set_fill(YELLOW, 1)
else:
new_row = self.get_guess_value_row(
self.guess_value_grid_lines[0], guess,
)
rows.shift(DOWN)
rows.add(new_row)
def get_shown_words(self, font_size=24):
return self.get_grid_of_words(
self.possibilities,
n_rows=20 - 2 * len(self.grid.words),
n_cols=1
)
def get_probability_bars(self, shown_words, max_width=1.0):
mobs = shown_words
words = shown_words.words
probs = [self.priors.get(w, 0) for w in words] # Unnormalized
height = mobs[0].get_height() * 0.7
bars = VGroup(*(
Rectangle(
width=prob * max_width,
height=height,
fill_color=self.prior_color,
fill_opacity=0.7,
stroke_width=0.5 * (prob > 0),
stroke_color=self.prior_color
)
for prob in probs
))
for bar, mob in zip(bars, mobs):
bar.next_to(mob, RIGHT, SMALL_BUFF)
bar.align_to(bars[0], LEFT)
# if not self.show_prior:
# bars.set_opacity(0)
return bars
def show_possible_words(self):
shown_words = self.get_shown_words()
count_label = self.get_count_label()
shown_words.next_to(count_label[:2], DOWN, buff=0.35)
prob_bars = self.get_probability_bars(shown_words)
if len(self.grid.words) > 0:
# Set up label transition
prev_count_label = self.count_label
count_label.shift(
(prev_count_label[1].get_right() - count_label[1].get_right())[0] * RIGHT
)
num_rate_func = squish_rate_func(rush_into, 0.3, 1)
def update_moving_count_label(label, alpha):
for i in (0, 2):
label[i].set_value(interpolate(
prev_count_label[i].get_value(),
count_label[i].get_value(),
num_rate_func(alpha),
))
label.set_y(interpolate(
prev_count_label.get_y(),
count_label.get_y(),
alpha
))
return label
label_transition = UpdateFromAlphaFunc(
prev_count_label.copy(),
update_moving_count_label,
remover=True
)
# Set up word transition
prev_words = self.shown_words
for shown_word, s_word in zip(shown_words, shown_words.words):
shown_word.save_state()
if s_word in prev_words.words:
index = prev_words.words.index(s_word)
shown_word.move_to(prev_words[index])
prev_words[index].set_opacity(0)
self.prob_bars[index].set_opacity(0)
elif "..." in prev_words.words:
shown_word.move_to(prev_words[prev_words.words.index("...")])
shown_word.set_opacity(0)
else:
shown_word.set_opacity(0)
prev_words.generate_target()
for i, word in enumerate(prev_words.words):
if word not in shown_words.words:
fader = prev_words.target[i]
fader.set_opacity(0)
fader.shift(LEFT)
# Set up bar transitions
for bar, s_word, word in zip(prob_bars, shown_words, shown_words.words):
bar.save_state()
if word not in prev_words.words:
bar.set_opacity(0)
bar.match_y(s_word)
bar.align_to(bar.saved_state, LEFT)
# Carry out animations
self.play(
FadeOut(self.prob_bars, run_time=0.25),
FadeOut(self.guess_value_grid, RIGHT),
label_transition,
MoveToTarget(prev_words, run_time=0.5),
LaggedStartMap(Restore, shown_words, run_time=1),
LaggedStartMap(Restore, prob_bars, run_time=1),
run_time=1,
)
self.add(count_label)
self.remove(prev_words)
shown_words.set_opacity(1)
self.add(count_label)
self.add(shown_words)
self.add(prob_bars)
self.count_label = count_label
self.shown_words = shown_words
self.prob_bars = prob_bars
def show_guess_values(self):
self.guess_value_grid = self.get_guess_value_grid()
self.play(ShowIncreasingSubsets(self.guess_value_grid))
def get_guess_value_grid(self, font_size=36):
if self.pre_computed_first_guesses and len(self.grid.words) == 0:
guesses = self.pre_computed_first_guesses
top_indices = np.arange(len(guesses))
else:
guesses = self.all_words
expected_scores = get_expected_scores(
guesses,
self.possibilities,
self.priors,
look_two_ahead=self.look_two_ahead
)
top_indices = np.argsort(expected_scores)[:self.n_top_picks]
guess_values_array = get_guess_values_array(
guesses,
self.possibilities,
self.priors,
look_two_ahead=self.look_two_ahead,
)
top_words = np.array(guesses)[top_indices]
top_guess_value_parts = guess_values_array[:, top_indices]
lines = self.get_guess_value_grid_lines()
guess_value_grid = VGroup(*(
self.get_guess_value_row(line, word, *values)
for line, word, values in zip(
lines, top_words, top_guess_value_parts.T
)
))
for value, row in zip(guess_values_array.sum(0)[top_indices], guess_value_grid):
if value == 0:
row.set_opacity(0)
guess_value_grid.set_stroke(background=True)
return guess_value_grid
def get_guess_value_row(self, line, word,
entropy=None,
entropy2=None,
probability=None,
font_size=36):
titles = self.guess_value_grid_titles
row = VGroup()
# Word
word_mob = Text(str(word), font="Consolas", font_size=font_size)
index = np.argmin([c.get_height() for c in word_mob])
aligner = word_mob[index]
word_mob.shift(line.get_center() - aligner.get_bottom() + 0.5 * SMALL_BUFF * UP)
word_mob.match_x(titles[0])
row.add(word_mob)
# Entropy
if entropy is None:
weights = get_weights(self.possibilities, self.priors)
entropy = get_entropies([word], self.possibilities, weights)[0]
dec_kw = dict(num_decimal_places=2, font_size=font_size)
row.add(DecimalNumber(entropy, color=self.entropy_color, **dec_kw))
# Second entropy
if self.look_two_ahead:
if entropy2 is None:
entropy2 = get_average_second_step_entropies(
[word], self.all_words, self.possibilities, self.priors
)
row.add(DecimalNumber(entropy2, color=self.entropy_color, **dec_kw))
# Prior
if self.show_prior:
if probability is None:
if word in self.possibilities:
weights = get_weights(self.possibilities, self.priors)
probability = weights[self.possibilities.index(word)]
else:
probability = 0
dec_kw['num_decimal_places'] = 5
# Dividing out by the weight given to prob in scores
row.add(DecimalNumber(probability, color=self.prior_color, **dec_kw))
for mob, title in zip(row, titles):
if mob is not word_mob:
mob.match_y(aligner, DOWN)
mob.match_x(title)
row.add(line)
return row
def get_guess_value_grid_lines(self):
titles = self.guess_value_grid_titles
line = Line().match_width(titles)
line.set_stroke(GREY_C, 1)
lines = line.get_grid(self.n_top_picks, 1, buff=0.5)
lines.next_to(titles, DOWN, buff=0.75)
return lines
def get_column_of_numbers(self, values, row_refs, col_ref, num_decimal_places=2, font_size=36):
mobs = VGroup(*(
DecimalNumber(
value,
num_decimal_places=num_decimal_places,
font_size=font_size
)
for value in values
))
for row_ref, mob in zip(row_refs, mobs):
mob.match_x(col_ref)
mob.match_y(row_ref)
return mobs
def on_key_press(self, symbol, modifiers):
if chr(symbol) == " ":
self.wait_to_proceed = False
super().on_key_press(symbol, modifiers)
class WordleDistributions(WordleScene):
grid_center = [-4.5, -0.5, 0]
grid_height = 5
bar_style = dict(
fill_color=TEAL_D,
fill_opacity=0.8,
stroke_color=WHITE,
stroke_width=0.1,
)
show_fraction_in_p_label = True
def get_axes(self,
x_max=3**5 / 2, y_max=0.1,
width=7.5,
height=6):
axes = Axes(
(0, x_max),
(0, y_max, y_max / 5),
height=height,
width=width,
x_axis_config={
"tick_size": 0,
}
)
axes.next_to(self.grid, RIGHT, LARGE_BUFF, aligned_edge=DOWN)
# y_label = OldTex("p(\\text{Pattern})", font_size=24)
# y_label.next_to(axes.y_axis.get_top(), UR, buff=SMALL_BUFF)
# axes.y_axis.add(y_label)
axes.y_axis.add_numbers(num_decimal_places=2)
x_label = Text("Pattern", font_size=24)
x_label.next_to(axes.x_axis.get_right(), DR, MED_SMALL_BUFF)
x_label.shift_onto_screen()
axes.x_axis.add(x_label)
self.axes = axes
return axes
def get_total_words_label(self, font_size=36):
label = VGroup(
Integer(len(self.all_words), font_size=font_size, edge_to_fix=UR),
Text("Total words", font_size=font_size)
)
label.arrange(RIGHT, aligned_edge=UP)
label.match_x(self.grid)
label.to_edge(UP, buff=MED_SMALL_BUFF)
return label
def get_dynamic_match_label(self, font_size=36):
label = VGroup(
Integer(len(self.possibilities), font_size=font_size, edge_to_fix=UR),
Text("Possible matches", font_size=font_size)
)
label.arrange(RIGHT, aligned_edge=DOWN)
label.set_max_width(self.grid.get_width())
label.next_to(self.grid, UP)
def update_label(label):
word = self.pending_word_as_string()
if self.grid.pending_pattern is None or not self.is_valid_guess():
label.set_opacity(0)
else:
buckets = get_word_buckets(word, self.possibilities)
bucket_size = len(buckets[self.grid.pending_pattern])
label[0].set_value(bucket_size)
label[0].next_to(label[1], LEFT, submobject_to_align=label[0][-1])
label.set_opacity(1)
label.add_updater(update_label)
return label
def get_bars(self, axes, values):
x_unit = axes.x_axis.unit_size
y_unit = axes.y_axis.unit_size
bars = Rectangle(width=x_unit, **self.bar_style).replicate(3**5)
for x, bar, value in zip(it.count(), bars, values):
bar.set_height(value * y_unit, stretch=True)
bar.move_to(axes.c2p(x, 0), DL)
return bars
def get_distribution_bars(self, axes, guess):
distribution = get_pattern_distributions(
[guess], self.possibilities,
get_weights(self.possibilities, self.priors)
)[0]
buckets = get_word_buckets(guess, self.possibilities)
pattern_indices = np.argsort(distribution)[::-1]
bars = self.get_bars(axes, distribution[pattern_indices])
bars.patterns = pattern_indices
for i, bar in enumerate(bars):
bar.prob = distribution[pattern_indices[i]]
bar.count = len(buckets[pattern_indices[i]])
return bars
def get_bar_indicator(self, bars, pattern_index):
pattern_index_tracker = ValueTracker(pattern_index)
def get_pattern_index():
return int(pattern_index_tracker.get_value())
def get_pattern():
return bars.patterns[get_pattern_index()]
tri = ArrowTip(angle=PI / 2)
tri.set_height(0.1)
tri.add_updater(lambda m: m.next_to(bars[get_pattern_index()], DOWN, buff=0))
row = self.get_curr_row()
row_copy = row.copy()
row_copy.scale(0.25)
row_copy.add_updater(lambda m: m.next_to(tri, DOWN, SMALL_BUFF))
bars.add_updater(lambda m: m.set_opacity(0.35))
bars.add_updater(lambda m: m[get_pattern_index()].set_opacity(1))
self.grid.add_updater(lambda m: self.show_pattern(get_pattern()))
self.add(self.grid)
row_copy.add_updater(lambda m: m.match_style(row).set_stroke(width=0.1))
self.mouse_drag_point.move_to(tri)
pattern_index_tracker.add_updater(lambda m: m.set_value(
clip(self.axes.x_axis.p2n(self.mouse_drag_point.get_center()), 0, 3**5)
))
indicator = Group(tri, row_copy)
def get_bar():
value = pattern_index_tracker.get_value()
index = int(clip(value, 0, 3**5 - 1))
return bars[index]
return indicator, pattern_index_tracker, get_bar
def get_dynamic_bar_label(self, tex, font_size=36):
row_copy = self.get_curr_row().copy()
row_copy.scale(0.25)
ndp = len(tex[-1].split(".")[1])
dec = DecimalNumber(0, num_decimal_places=ndp, font_size=font_size)
result = VGroup(*Tex(*tex, font_size=font_size))
row_copy.replace(result[1], dim_to_match=0)
dec.replace(result[-1])
result.replace_submobject(1, row_copy)
result.remove(result[-1])
result.add(dec)
result.add_updater(lambda m: m[1].match_style(self.get_curr_row()).set_stroke(WHITE, 0.1))
return result
def get_p_label(self, get_bar, max_y=1):
poss_string = "{:,}".format(len(self.possibilities)).replace(",", "{,}")
strs = ["p\\left(", "00000", "\\right)", "="]
if self.show_fraction_in_p_label:
strs.extend(["{" + poss_string, "\\over ", poss_string + "}", "=", ])
strs.append("0.0000")
p_label = self.get_dynamic_bar_label(strs)
if self.show_fraction_in_p_label:
num = Integer(edge_to_fix=DOWN, font_size=36)
num.move_to(p_label[4], DOWN)
p_label.replace_submobject(4, num)
def update_label(label):
label[4].set_value(int(get_bar().count))
label[-1].set_value(get_bar().prob)
label.next_to(get_bar(), UR, SMALL_BUFF)
label.set_y(min(max_y, label.get_y()))
label.shift_onto_screen(buff=1.0)
p_label.add_updater(update_label)
return p_label
def get_information_label(self, p_label, get_bar):
info_label = self.get_dynamic_bar_label(
(
"I\\left(", "00000", "\\right)", "=",
"\\log_2\\left(1 / p)", "=",
"0.00"
),
)
info_label.add_updater(lambda m: m[-1].set_value(-safe_log2(get_bar().prob)))
info_label.add_updater(lambda m: m.next_to(p_label, UP, aligned_edge=LEFT))
info_label.add_updater(lambda m: m.shift_onto_screen())
return info_label
def get_entropy_label(self, font_size=36):
guess = self.pending_word_as_string()
if self.is_valid_guess():
entropy = get_entropies(
[guess],
self.possibilities,
get_weights(self.possibilities, self.priors)
)[0]
else:
entropy = 0
lhs = OldTex(
"E[I] = \\sum_x ",
"p(x) \\cdot \\log_2(1 / p(x))", "=",
tex_to_color_map={"I": BLUE},
font_size=font_size,
)
value = DecimalNumber(entropy, font_size=font_size)
value.next_to(lhs[-1], RIGHT)
result = VGroup(lhs, value)
result.move_to(self.axes, UR)
result.to_edge(UP)
return result
def get_grid_of_matches(self, n_rows=20, n_cols=9):
if self.grid.pending_pattern is not None:
buckets = get_word_buckets(
self.pending_word_as_string(),
self.possibilities
)
words = buckets[self.grid.pending_pattern]
else:
words = self.possibilities
word_mobs = self.get_grid_of_words(words, n_rows, n_cols)
word_mobs.move_to(midpoint(self.grid.get_right(), RIGHT_SIDE))
return word_mobs
# Animations
def add_distribution(self, axes):
pass
class ExternalPatternEntry(WordleSceneWithAnalysis):
# uniform_prior = True
# wordle_based_prior = True
def setup(self):
self.pending_pattern = []
super().setup()
def get_pattern(self, guess):
if len(self.pending_pattern) == 5:
return pattern_from_string(self.pending_pattern)
return None
def reveal_pattern(self, *args, **kwargs):
super().reveal_pattern(*args, **kwargs)
self.pending_pattern = []
# Interactive parts
def on_key_press(self, symbol, modifiers):
char = chr(symbol)
if '0' <= char <= '2' and len(self.pending_pattern) < 5:
square = self.get_curr_row()[len(self.pending_pattern)]
square.set_fill(self.color_map[int(char)], 1)
self.pending_pattern.append(char)
super().on_key_press(symbol, modifiers)
class TitleCardScene(WordleScene):
grid_height = 7
words = ["three", "blues", "wonts"]
secret_word = "brown"
reveal_run_time = 1
reveal_lag_ratio = 0.2
def construct(self):
for guess in self.words:
for letter in guess:
self.add_letter(letter)
self.wait(0.05 + random.random() * 0.15)
self.reveal_pattern()
self.wait(0.25)
self.wait()
def is_valid_guess(self):
# Everyone's a winner!
return True
def refresh_possibilities(self, guess, pattern):
pass
# Scenes
class LessonTitleCard(Scene):
def construct(self):
title = VGroup(
Text("Lesson today:", font_size=40),
Text("Information theory", color=TEAL),
)
title.arrange(DOWN)
title.scale(1.5)
title.center()
title[1].add(
Underline(title[1], buff=-0.05).set_stroke(GREY, 2)
)
self.add(title[0])
self.play(Write(title[1], run_time=1))
self.wait()
self.play(title.animate.scale(1 / 1.5).to_edge(UP))
self.wait()
class DrawPhone(Scene):
def construct(self):
morty = Mortimer().flip()
morty.center().to_edge(DOWN)
phone = SVGMobject("wordle-phone")
phone.set_height(5)
phone.next_to(morty.get_corner(UR), RIGHT)
phone.shift(UP)
phone.set_fill(opacity=0)
phone.set_stroke(WHITE, 0)
bubble = ThoughtBubble(height=4, width=4, direction=RIGHT)
bubble.next_to(morty, UL, buff=0)
bubble.add_content(WordleScene.patterns_to_squares(
list(map(pattern_from_string, ["00102", "00110", "22222"]))
))
bubble.content.scale(0.7)
self.add(morty)
self.add(phone)
self.play(
LaggedStart(
morty.change("thinking", phone),
ShowCreation(bubble),
FadeIn(bubble.content, lag_ratio=0.1)
),
Write(phone, run_time=3, lag_ratio=0.01),
)
self.play(
Blink(morty),
FadeOut(phone),
)
self.wait()
class AskWhatWorldeIs(TeacherStudentsScene):
def construct(self):
self.student_says(
"What is Wordle?",
index=0,
target_mode="raise_left_hand",
added_anims=[
self.teacher.change("tease")
]
)
self.play_student_changes(
"raise_left_hand", "hesitant", "happy",
look_at=self.students[0].bubble,
)
self.wait(3)
class IntroduceGame(WordleScene):
secret_word = "brown"
grid_center = 3.5 * LEFT
def construct(self):
# Secret
grid = self.grid
row_copy = self.get_curr_row().copy()
secret = VGroup(
Text("Secret word"),
Vector(0.5 * DOWN),
row_copy,
)
secret[2].match_width(secret[0])
secret.arrange(DOWN, buff=MED_SMALL_BUFF)
secret.next_to(grid, RIGHT, buff=2.0, aligned_edge=UP)
word_list = random.sample(get_word_list(short=True), 100)
word_list.append("?????")
words = VGroup(*(Text(word, font="Consolas") for word in word_list))
words.set_height(row_copy.get_height() * 0.7)
words[-1].set_color(RED)
for word in words:
for char, square in zip(word, row_copy):
char.move_to(square)
self.wait()
self.wait()
self.play(
Write(secret[0]),
ShowCreation(secret[1]),
TransformFromCopy(grid[0], secret[2])
)
self.play(
ShowSubmobjectsOneByOne(words, run_time=10, rate_func=linear),
)
self.wait()
self.row_copy = row_copy
self.q_marks = words[-1]
# Guesses
numbers = VGroup(*(Integer(i) for i in range(1, 7)))
for number, row in zip(numbers, grid):
number.next_to(row, LEFT)
self.play(FadeIn(numbers, lag_ratio=0.1))
guesses = ["three", "blues", "one", "onnne", "wonky", "brown"]
if not self.presenter_mode:
for guess in guesses:
self.add_word(guess)
self.reveal_pattern()
else:
self.wait(note=f"Type {guesses}")
# Show word lists
all_words = get_word_list()
answers = get_word_list(short=True)
titles = VGroup(
VGroup(Text("Allowed guesses"), Integer(len(all_words))),
VGroup(Text("Possible answers"), Integer(len(answers))),
)
for title in titles:
title.arrange(DOWN)
titles.scale(0.8)
titles.arrange(RIGHT, buff=1.5, aligned_edge=UP)
titles[1][1].match_y(titles[0][1])
titles.to_corner(UR)
titles.to_edge(RIGHT, buff=MED_LARGE_BUFF)
word_columns = VGroup(
self.get_grid_of_words(all_words, 20, 5),
self.get_grid_of_words(answers, 20, 1),
)
word_columns[1].set_color(GREEN)
for column, title in zip(word_columns, titles):
column.next_to(title, DOWN, MED_LARGE_BUFF)
grid.add(numbers)
self.play(
FadeOut(secret),
FadeOut(self.q_marks),
grid.animate.scale(0.7, about_edge=LEFT),
Write(titles[0][0], run_time=1),
)
self.play(
CountInFrom(titles[0][1], 0),
ShowIncreasingSubsets(word_columns[0]),
)
self.wait()
self.play(
FadeIn(titles[1][0]),
CountInFrom(titles[1][1], 0),
ShowIncreasingSubsets(word_columns[1]),
)
self.wait()
# Try not to use wordle_words
frame = self.camera.frame
answer_rect = SurroundingRectangle(VGroup(titles[1], word_columns[1]))
answer_rect.set_stroke(TEAL, 3)
avoid = OldTexText("Let's try to avoid\\\\using this")
avoid.next_to(answer_rect, RIGHT)
morty = Mortimer(height=2)
morty.next_to(avoid, DOWN, MED_LARGE_BUFF)
morty.change("hesitant", answer_rect)
morty.save_state()
morty.change("plain").set_opacity(0)
self.play(
frame.animate.match_x(word_columns, LEFT).shift(LEFT),
ShowCreation(answer_rect),
run_time=2,
)
self.play(
Write(avoid),
Restore(morty),
)
self.wait()
# Common but not in wordle list
priors = get_frequency_based_priors()
not_in_answers = set(all_words).difference(answers)
sorted_by_freq = list(sorted(not_in_answers, key=lambda w: priors[w]))
n = 15
most_common = self.get_grid_of_words(sorted_by_freq[-n:], n, 1)
most_common.set_color(BLUE)
most_common.move_to(morty.get_corner(UR), DOWN).shift(MED_SMALL_BUFF * UP)
non_s_most_common = self.get_grid_of_words(
list(filter(lambda w: w[-1] != 's', sorted_by_freq))[-n:], n, 1
)
non_s_most_common.match_style(most_common)
non_s_most_common.replace(most_common)
label = Text("Not in wordle list", font_size=36)
label.next_to(most_common, UP)
self.play(
FadeOut(avoid, DOWN),
morty.change("raise_LEFT_hand", most_common),
ShowIncreasingSubsets(most_common),
)
self.play(FadeIn(label))
self.play(Blink(morty))
self.wait()
self.play(
morty.change("pondering", most_common),
LaggedStartMap(FadeOut, most_common, shift=RIGHT),
LaggedStartMap(FadeIn, non_s_most_common, shift=RIGHT),
)
self.wait()
def show_pattern(self, *args, **kwargs):
guess = self.pending_word_as_string()
letters = self.grid.pending_word.copy()
for letter, q_mark in zip(letters, self.q_marks):
letter.replace(q_mark, dim_to_match=1)
added_anims = []
if guess == self.secret_word:
added_anims.append(LaggedStart(
*(
square.animate.set_fill(GREEN, 1)
for square in self.row_copy
),
lag_ratio=0.7,
run_time=2
))
added_anims.append(LaggedStart(
*(
Transform(q_mark, letter)
for q_mark, letter in zip(self.q_marks, letters)
),
lag_ratio=0.7,
run_time=2
))
super().show_pattern(*args, added_anims=added_anims, **kwargs)
class InitialDemo(ExternalPatternEntry):
secret_word = "elder"
pre_computed_first_guesses = [
"crane", "slane", "slate", "salet", "trace",
"reast", "crate", "toile", "torse", "carse",
"carle", "trone", "carte", "roast",
]
# wordle_based_prior = True
class ShowTonsOfWords(Scene):
def construct(self):
words = get_word_list()
n_rows = 18
n_cols = 15
N = n_rows * n_cols
grids = VGroup(*(
WordleScene.get_grid_of_words(words[N * k:N * (k + 1)], n_rows, n_cols)
for k in range(5)
))
grids.set_width(FRAME_WIDTH - 3)
grids.arrange(DOWN, buff=0.8 * SMALL_BUFF)
grids.to_edge(UP)
self.add(grids)
frame = self.camera.frame
self.play(frame.animate.move_to(grids, DOWN), run_time=15)
class FinalPerformanceFrame(VideoWrapper):
title = "Final performance"
animate_boundary = False
class FirstThoughtsTitleCard(TitleCardScene):
n_letters = 5
words = ["first", "naive", "ideas"]
secret_word = "start"
class ChoosingBasedOnLetterFrequencies(IntroduceGame):
def construct(self):
# Reconfigure grid to be flat
grid = self.grid
grid.set_submobjects([VGroup(*it.chain(*grid))])
grid.add(grid.pending_word)
self.add(grid)
# Data on right
letters_and_frequencies = [
("E", 13),
("T", 9.1),
("A", 8.2),
("O", 7.5),
("I", 7),
("N", 6.7),
("S", 6.3),
("H", 6.1),
("R", 6),
("D", 4.3),
("L", 4),
("U", 2.8),
("C", 2.8),
("M", 2.5),
("W", 2.4),
("F", 2.2),
("G", 2.0),
("Y", 2.0),
]
freq_data = VGroup(*(
VGroup(
Text(letter, font="Consolas"),
Rectangle(
height=0.25, width=0.2 * freq,
stroke_width=0,
fill_color=(BLUE if letter in "AEIOUY" else GREY_B),
fill_opacity=1,
),
DecimalNumber(freq, num_decimal_places=1, font_size=24, unit="\\%")
).arrange(RIGHT)
for letter, freq in letters_and_frequencies
))
freq_data.arrange(DOWN, aligned_edge=LEFT)
freq_data.set_height(FRAME_HEIGHT - 1)
freq_data.to_edge(RIGHT, buff=LARGE_BUFF)
self.freq_data = freq_data
for row, lf in zip(freq_data, letters_and_frequencies):
letter = lf[0]
row.letter = letter
row.rect = SurroundingRectangle(row, buff=SMALL_BUFF)
row.rect.set_stroke(YELLOW, 0)
row.add(row.rect)
freq_data.add_updater(lambda m: m)
self.add(freq_data)
def add_letter(self, letter):
super().add_letter(letter)
self.update_freq_data_highlights()
def delete_letter(self):
super().delete_letter()
self.update_freq_data_highlights()
def update_freq_data_highlights(self):
word = self.pending_word_as_string()
for row in self.freq_data:
if row.letter.lower() in word.lower():
row.set_opacity(1)
row.rect.set_fill(opacity=0)
row.rect.set_stroke(width=2)
else:
row.set_opacity(0.5)
row.rect.set_fill(opacity=0)
row.rect.set_stroke(width=0)
class ExampleGridColors(WordleScene):
grid_center = ChoosingBasedOnLetterFrequencies.grid_center
secret_word = "baker"
def construct(self):
self.wait(3)
grid = self.grid
for guess in ["other", "nails"]:
self.add_word(guess, 0)
self.reveal_pattern()
self.wait()
for color in [BLACK, self.color_map[0]]:
grid.generate_target()
grid.target[:2].set_fill(color, 1),
self.play(
MoveToTarget(grid),
lag_ratio=0.5,
run_time=2
)
self.wait()
self.embed()
class PreviewGamePlay(WordleSceneWithAnalysis):
n_games = 10
pre_computed_first_guesses = [
"tares", "lares", "rates", "rales", "tears",
"tales", "salet", "teras", "arles", "nares",
"soare", "saner", "reals"
]
def construct(self):
self.show_guess_values()
self.initial_guess_value_grid = self.guess_value_grid
for x in range(self.n_games):
self.clear()
self.setup()
self.secret_word = random.choice(get_word_list(short=True))
self.guess_value_grid = self.initial_guess_value_grid
self.add(self.guess_value_grid)
while not self.has_won():
guess = self.guess_value_grid[0][0].text
self.add_word(guess)
self.wait(0.5)
self.reveal_pattern()
class UlteriorMotiveWrapper(VideoWrapper):
title = "Ulterior motive: Lesson on entropy"
class IntroduceDistribution(WordleDistributions):
secret_word = "brown"
uniform_prior = True
n_word_rows = 20
n_bars_to_analyze = 3
def construct(self):
# Total labels
total_label = self.get_total_words_label()
self.add(total_label)
# Show an example guess
guess = "weary"
match_label = self.get_dynamic_match_label()
word_grid = self.get_grid_of_matches(n_rows=self.n_word_rows)
self.wait()
self.wait()
self.play(ShowIncreasingSubsets(word_grid, run_time=3))
self.wait(note=f"Write {guess}, (but don't submit)")
if not self.presenter_mode or self.skip_animations:
self.add_word(guess)
# Show several possible patterns, with corresponding matches
pattern_strs = ["20100", "01000"]
prob_label = VGroup()
for i, pattern_str in enumerate(pattern_strs):
pattern = pattern_from_string(pattern_str)
self.remove(match_label)
self.show_pattern(pattern, animate=True)
self.play(FadeOut(word_grid), FadeOut(prob_label))
word_grid = self.get_grid_of_matches(n_rows=self.n_word_rows)
self.add(match_label)
match_label.update()
self.play(
CountInFrom(match_label[0], 0),
ShowIncreasingSubsets(word_grid, run_time=2)
)
self.wait(note=f"Pattern {i} / {len(pattern_strs)}")
num = match_label[0].get_value()
denom = total_label[0].get_value()
prob_label = self.get_dynamic_bar_label((
"p\\left(", "0000", "\\right)", "=",
"{" + "{:,}".format(num).replace(",", "{,}"), "\\over ",
"{:,}".format(denom).replace(",", "{,}") + "}", "=",
"0.0000",
))
prob_label[-1].set_value(num / denom)
prob_label.next_to(word_grid, UP)
prob_label.clear_updaters()
self.play(
LaggedStart(
FadeTransform(match_label[0].copy().clear_updaters(), prob_label[4], remover=True),
FadeTransform(total_label[0].copy().clear_updaters(), prob_label[6], remover=True),
lag_ratio=0.5,
),
FadeIn(VGroup(*prob_label[:4], prob_label[5], prob_label[7:])),
)
self.add(prob_label)
self.wait()
# Show distribution
axes = self.get_axes(y_max=0.15)
bars = self.get_distribution_bars(axes, guess)
self.play(
FadeOut(word_grid),
FadeOut(prob_label),
)
self.play(LaggedStart(
Write(axes),
ShowIncreasingSubsets(bars),
lag_ratio=0.5
))
self.add(bars)
self.wait()
index = 20
bar_indicator, x_tracker, get_bar = self.get_bar_indicator(bars, index)
p_label = self.get_p_label(get_bar)
self.add(bar_indicator, x_tracker)
self.add(p_label)
self.add(match_label)
for x in range(self.n_bars_to_analyze):
self.wait(note=f"Play around with probability {x} / {self.n_bars_to_analyze}")
word_grid = self.get_grid_of_matches(n_rows=12, n_cols=5)
word_grid.next_to(p_label, UP, LARGE_BUFF)
self.play(ShowIncreasingSubsets(word_grid))
self.wait()
self.remove(word_grid)
# Describe aim for expected information
want = Text("What we want:")
standin = OldTex(
"E[\\text{Information}] = \\sum_{x} p(x) \\cdot (\\text{Something})",
tex_to_color_map={
"\\text{Something}": GREY_B,
"\\text{Information}": BLUE,
},
font_size=36,
)
group = VGroup(want, standin)
group.arrange(DOWN, buff=MED_LARGE_BUFF)
group.to_corner(UR)
self.play(FadeIn(want))
self.play(Write(standin))
self.wait(note="Discuss adding up over all patterns")
# Define information
info_label = self.get_information_label(p_label, get_bar)
il_copy = info_label.copy().clear_updaters()
self.play(FadeIn(il_copy, UP, remover=True))
self.add(info_label)
self.wait(note="Give intuitions on values of I")
# Define entropy
entropy_definition = self.get_entropy_label()
brace = Brace(entropy_definition, DOWN, buff=SMALL_BUFF)
ent_label = OldTexText("Entropy, $H$")
ent_label.set_color(BLUE)
ent_label.next_to(brace, DOWN, SMALL_BUFF)
self.play(
FadeIn(entropy_definition, UP),
FadeOut(standin, UP),
FadeOut(want, UP),
)
self.wait(note="Drag tracker through full distributinon")
self.wait()
self.play(x_tracker.animate.set_value(10), run_time=3)
self.play(
GrowFromCenter(brace),
Write(ent_label)
)
self.wait()
# Show an alternate word
self.remove(bar_indicator, x_tracker, p_label, info_label, bars, match_label)
entropy_definition[-1].set_opacity(0)
self.grid.clear_updaters()
self.grid.add_updater(lambda m: m)
self.get_curr_row().set_fill(BLACK, 0)
self.grid.pending_pattern = None
self.wait(note="Delete word, write \"saner\", but don't enter!")
if not self.presenter_mode or self.skip_animations:
for x in range(5):
self.delete_letter()
self.add_word('saner')
guess = self.pending_word_as_string()
bars = self.get_distribution_bars(axes, guess)
self.play(ShowIncreasingSubsets(bars, run_time=3))
self.wait()
trackers = self.add_trackers(bars, index=20)
self.wait(note="Play around more with distribiution")
self.recalculate_entropy(entropy_definition, guess, trackers[0])
self.wait()
# Examples of good entropy
self.remove(*trackers)
bars.set_opacity(0.7)
self.grid.clear_updaters()
self.grid.add_updater(lambda m: m)
self.grid.pending_word.set_submobjects([])
self.add_word("?????")
self.get_curr_row().set_fill(BLACK, 0)
eqs = VGroup(
OldTex("E[I] = \\log_2\\left({1 \\over 1 / 3^5}\\right) = \\log_2(3^5) \\approx 7.92"),
OldTex("E[I] = \\log_2\\left({1 \\over 1 / 16}\\right) = \\log_2(16) = 4.00"),
OldTex("E[I] = \\log_2\\left({1 \\over 1 / 64}\\right) = \\log_2(64) = 6.00"),
)
eqs.scale(0.7)
for eq in eqs:
eq.next_to(ent_label, DOWN, LARGE_BUFF)
prev_values = [bar.prob for bar in bars]
last_eq = VGroup()
ent_rhs = entropy_definition[-1]
for eq, x in zip(eqs, [3**5, 16, 64]):
values = [1 / x] * x + [0] * (3**5 - x)
self.set_bars_to_values(bars, values, ent_rhs, added_anims=[FadeOut(last_eq)])
self.wait()
self.play(FadeIn(eq, UP))
last_eq = eq
self.wait()
self.grid.pending_word.set_submobjects([])
self.add_word(guess, wait_time_per_letter=0)
self.set_bars_to_values(bars, prev_values, ent_rhs, added_anims=[FadeOut(last_eq)])
self.wait()
# Show the second guess
true_pattern = self.get_pattern(guess)
self.show_pattern(true_pattern, animate=True)
trackers[0].set_value(list(bars.patterns).index(true_pattern))
match_label.update()
self.play(FadeIn(match_label))
self.wait()
self.play(
ApplyMethod(
match_label[0].copy().move_to,
total_label[0], UR,
remover=True,
),
FadeOut(total_label[0], UP)
)
total_label[0].set_value(match_label[0].get_value())
self.add(total_label)
self.wait()
faders = [axes, bars, match_label]
for fader in faders:
fader.clear_updaters()
self.play(
LaggedStart(*map(FadeOut, faders)),
FadeOut(entropy_definition[-1]),
)
self.grid.clear_updaters()
self.reveal_pattern(animate=False)
next_guess = "wordy"
self.wait(note=f"Type in \"{next_guess}\"")
if not self.presenter_mode or self.skip_animations:
self.add_word(next_guess)
# Show new distribution
guess = self.pending_word_as_string()
axes = self.get_axes(y_max=1, x_max=50)
bars = self.get_distribution_bars(axes, guess)
self.play(
FadeIn(axes),
FadeIn(bars, lag_ratio=0.1, run_time=2)
)
self.wait()
self.remove(match_label)
trackers = self.add_trackers(bars)
self.wait(note="Play around with distribution")
trackers[0].clear_updaters()
self.play(trackers[0].animate.set_value(10), run_time=3)
self.play(trackers[0].animate.set_value(0), run_time=5)
self.recalculate_entropy(entropy_definition, guess)
# Other second guesses
self.grid.clear_updaters()
self.get_curr_row().set_fill(BLACK, 0)
self.pending_pattern = None
entropy_definition[1].set_opacity(0)
self.remove(bars, *trackers)
if not self.presenter_mode or self.skip_animations:
words = get_word_list()
for guess in random.sample(words, 15):
for x in range(5):
self.delete_letter()
self.add_word(guess, wait_time_per_letter=0)
bars = self.get_distribution_bars(axes, guess)
self.add(bars)
self.play(FadeIn(bars, lag_ratio=0.1, run_time=2))
self.recalculate_entropy(entropy_definition, guess)
self.wait()
self.remove(bars)
else:
guess = "print"
self.wait(note=f"Write \"{guess}\"")
trackers = self.add_trackers(bars)
self.wait(note="Play around with distribution")
def set_bars_to_values(self, bars, values, ent_rhs, run_time=3, added_anims=[]):
y_unit = self.axes.y_axis.unit_size
bars.generate_target()
bar_template = bars[0].copy()
entropy = entropy_of_distributions(np.array(values))
for bar, value in zip(bars.target, values):
target = bar_template.copy()
target.set_height(
y_unit * value,
stretch=True,
about_edge=DOWN
)
target.move_to(bar, DOWN)
bar.become(target)
self.play(
MoveToTarget(bars, run_time=run_time),
ChangeDecimalToValue(ent_rhs, entropy, run_time=run_time),
*added_anims
)
def recalculate_entropy(self, entropy_definition, guess, x_tracker=None):
dec = entropy_definition[-1]
dec.set_value(
get_entropies(
[guess], self.possibilities,
get_weights(self.possibilities, self.priors)
)[0]
)
dec.set_opacity(1)
dec.next_to(entropy_definition[-2][-1])
anims = [CountInFrom(dec, 0)]
run_time = 1
if x_tracker is not None:
x_tracker.suspend_updating()
anims.append(UpdateFromAlphaFunc(
x_tracker, lambda m, a: m.set_value(a * 200),
run_time=5,
))
run_time = 3
self.play(*anims, run_time=run_time)
self.wait()
def add_trackers(self, bars, index=1):
bar_indicator, x_tracker, get_bar = self.get_bar_indicator(bars, index)
p_label = self.get_p_label(get_bar, max_y=1)
info_label = self.get_information_label(p_label, get_bar)
match_label = self.get_dynamic_match_label()
trackers = [x_tracker, bar_indicator, p_label, info_label, match_label]
self.add(*trackers)
return trackers
class ButTheyreNotEquallyLikely(Scene):
def construct(self):
randy = Randolph()
morty = Mortimer()
pis = VGroup(randy, morty)
pis.arrange(RIGHT, buff=2)
pis.to_edge(DOWN)
randy.make_eye_contact(morty)
self.play(
PiCreatureSays(
randy, OldTexText("But they're \\emph{not}\\\\equally likely!"),
target_mode="angry",
),
morty.change("guilty", randy.eyes),
)
self.play(Blink(randy))
self.wait()
self.play(
PiCreatureSays(
morty, OldTexText("To warm up, let's\\\\assume they are."),
target_mode="speaking",
),
RemovePiCreatureBubble(randy),
)
self.play(Blink(morty))
self.wait()
class KeyIdea(Scene):
def construct(self):
title = Text("Key idea", font_size=72)
title.to_edge(UP, LARGE_BUFF)
title.add(Underline(title, buff=-SMALL_BUFF, stroke_width=0.5).scale(1.5))
title.set_color(YELLOW)
idea = OldTexText("Informative", " $\\Leftrightarrow$", " Unlikely", font_size=72)
self.add(title)
idea[0].save_state()
idea[0].center()
self.play(FadeIn(idea[0]))
self.wait()
self.play(
Restore(idea[0]),
FadeIn(idea[1], RIGHT),
FadeIn(idea[2], 2 * RIGHT)
)
self.play(FlashAround(idea, run_time=2))
self.wait()
class ExpectedMatchesInsert(Scene):
def construct(self):
tex = OldTex(
"\\sum_{x} p(x) \\big(\\text{\\# Matches}\\big)",
tex_to_color_map={"\\text{\\# Matches}": GREEN}
)
cross = Cross(tex).scale(1.25)
self.play(Write(tex))
self.wait()
self.play(ShowCreation(cross))
self.wait()
class InformationTheoryTitleCard(TitleCardScene):
n_letters = 6
words = ["inform", "ation-", "theory", "basics"]
class DescribeBit(TeacherStudentsScene):
def construct(self):
words = OldTexText(
"Standard unit of\\\\information: The bit",
tex_to_color_map={"The bit": YELLOW}
)
words.next_to(self.teacher.get_corner(UL), UP, MED_LARGE_BUFF)
words.to_edge(RIGHT)
self.play(
self.teacher.change("raise_left_hand"),
FadeIn(words, UP)
)
self.play_student_changes(
"happy", "pondering", "thinking",
look_at=words
)
self.wait(4)
formula = OldTex("I = -\\log_2(p)", tex_to_color_map={"I": YELLOW})
formula.next_to(self.teacher.get_corner(UL), UP)
self.play(
words.animate.to_edge(UP),
FadeIn(formula, UP),
self.teacher.change("raise_right_hand", formula),
self.students[0].change("erm", formula),
self.students[1].change("confused", formula),
self.students[2].change("pondering", formula),
)
self.wait(5)
class DefineInformation(Scene):
def construct(self):
# Spaces
pre_space = Square(side_length=3)
pre_space.set_stroke(WHITE, 2)
pre_space.set_fill(BLUE, 0.7)
post_space = self.get_post_space(pre_space, 1)
arrow = Vector(2 * RIGHT)
group = VGroup(pre_space, arrow, post_space)
group.arrange(RIGHT)
# Labels
kw = dict(font_size=36)
pre_label = Text("Space of possibilities", **kw)
pre_label.next_to(pre_space, UP, SMALL_BUFF)
obs_label = Text("Observation", **kw)
obs_label.next_to(arrow, UP)
post_labels = self.get_post_space_labels(post_space, **kw)
# 1 bit (has an s)
self.add(pre_space)
self.add(pre_label)
self.wait()
self.wait()
self.play(
ShowCreation(arrow),
FadeIn(obs_label, lag_ratio=0.1),
FadeTransform(pre_space.copy().set_fill(opacity=0), post_space),
FadeIn(post_labels[1], 3 * RIGHT),
)
self.wait()
self.play(Write(post_labels[0], run_time=1))
self.wait()
# Show all words
n_rows = 25
n_cols = 8
all_words = get_word_list()
words_sample = random.sample(all_words, n_rows * n_cols)
word_grid = WordleScene.get_grid_of_words(words_sample, n_rows, n_cols)
word_grid.replace(pre_space, dim_to_match=1)
word_grid.scale(0.95)
word_grid.shuffle()
for word in word_grid:
word.save_state()
word_grid.scale(2)
word_grid.set_opacity(0)
self.play(LaggedStartMap(Restore, word_grid, lag_ratio=0.02, run_time=2))
self.wait()
word_grid.save_state()
word_grid.generate_target()
for word in word_grid.target:
if 's' not in word.text:
word.set_opacity(0.1)
has_s = OldTexText("Has an `s'", font_size=24)
has_s.next_to(arrow, DOWN)
self.play(
MoveToTarget(word_grid),
FadeIn(has_s, 0.5 * DOWN),
)
self.wait()
# 2 bits (has a t)
frame = self.camera.frame
mini_group1 = self.get_mini_group(pre_space, arrow, post_space, post_labels)
mini_group1.target.to_edge(UP, buff=0.25)
post_space2 = self.get_post_space(pre_space, 2).move_to(post_space)
post_labels2 = self.get_post_space_labels(post_space2, **kw)
has_t = OldTexText("Has a `t'", font_size=24)
has_t.next_to(arrow, DOWN, SMALL_BUFF)
self.play(
MoveToTarget(mini_group1),
FadeOut(has_s),
Restore(word_grid),
FadeOut(post_space),
FadeOut(post_labels),
frame.animate.move_to(2 * RIGHT)
)
self.play(
FadeTransform(pre_space.copy(), post_space2),
FadeIn(post_labels2, shift=3 * RIGHT)
)
self.wait()
word_grid.generate_target()
for word in word_grid.target:
if 't' not in word.text:
word.set_opacity(0.1)
self.play(
FadeIn(has_t, 0.5 * DOWN),
MoveToTarget(word_grid),
)
self.wait()
self.remove(has_t)
word_grid.restore()
# 3 through 5 bits
last_posts = VGroup(post_space2, post_labels2)
mini_groups = VGroup(mini_group1)
for n in range(3, 7):
new_mini = self.get_mini_group(pre_space, arrow, *last_posts)
new_mini.target.next_to(mini_groups, DOWN, buff=0.5)
new_post_space = self.get_post_space(pre_space, n)
new_post_space.move_to(post_space)
new_post_labels = self.get_post_space_labels(new_post_space, **kw)
self.play(LaggedStart(
MoveToTarget(new_mini),
AnimationGroup(
FadeOut(last_posts),
FadeIn(new_post_space),
FadeIn(new_post_labels),
),
lag_ratio=0.5
))
self.wait()
mini_groups.add(new_mini)
last_posts = VGroup(new_post_space, new_post_labels)
# Show formula
group = VGroup(pre_space, pre_label, word_grid, arrow, obs_label, *last_posts)
kw = dict(tex_to_color_map={"I": YELLOW})
formulas = VGroup(
OldTex("\\left( \\frac{1}{2} \\right)^I = p", **kw),
OldTex("2^I = \\frac{1}{p}", **kw),
OldTex("I = \\log_2\\left(\\frac{1}{p}\\right)", **kw),
OldTex("I = -\\log_2(p)", **kw)
)
formulas[:3].arrange(RIGHT, buff=LARGE_BUFF)
formulas[:3].to_edge(UP)
formulas[1:3].match_y(formulas[0][-1][0])
formulas[3].next_to(formulas[2], DOWN, aligned_edge=LEFT)
formulas[0].save_state()
formulas[0].move_to(formulas[1])
self.play(
FadeIn(formulas[0]),
group.animate.to_edge(DOWN, buff=MED_SMALL_BUFF)
)
self.wait()
self.play(Restore(formulas[0]))
for i in (0, 1, 2):
self.play(TransformMatchingShapes(formulas[i].copy(), formulas[i + 1]))
self.wait()
rhs_rect = SurroundingRectangle(formulas[3])
rhs_rect.set_stroke(YELLOW, 2)
self.play(ShowCreation(rhs_rect))
self.wait()
# Ask why?
randy = Randolph("confused", height=1.5)
randy.next_to(formulas[2], DL, MED_LARGE_BUFF)
randy.look_at(rhs_rect)
randy.save_state()
randy.change("plain")
randy.set_opacity(0)
self.play(Restore(randy))
self.play(Blink(randy))
self.wait()
self.play(randy.change("maybe"))
self.play(Blink(randy))
self.wait()
# Readibility
expr = OldTex(
"20 \\text{ bits} \\Leftrightarrow p \\approx 0.00000095",
tex_to_color_map={"\\text{bits}": YELLOW}
)
expr.next_to(group, UP, buff=0.75)
self.play(
FadeOut(randy),
FadeOut(formulas[3]),
FadeOut(rhs_rect),
Write(expr),
)
self.wait()
# Additive
group = group[:-2]
self.play(
FadeOut(expr),
FadeOut(last_posts),
group.animate.scale(0.7, about_edge=DL),
FadeOut(mini_groups, RIGHT),
frame.animate.move_to(RIGHT),
)
ps1 = self.get_post_space(pre_space, 2)
ps2 = self.get_post_space(pre_space, 5)
ps2.set_stroke(width=1)
ps2.add(ps1.copy().fade(0.5))
arrow2 = arrow.copy()
ps1.next_to(arrow, RIGHT)
arrow2.next_to(ps1, RIGHT)
ps2.next_to(arrow2, RIGHT)
ps1.label = self.get_post_space_labels(ps1, font_size=24)
ps2.label = self.get_post_space_labels(
self.get_post_space(pre_space, 3).replace(ps2),
font_size=24
)
self.play(
FadeTransform(pre_space.copy().set_opacity(0), ps1),
FadeIn(ps1.label, 2 * RIGHT),
)
self.wait()
self.play(
FadeTransform(ps1.copy().set_opacity(0), ps2),
FadeIn(ps2.label, 2 * RIGHT),
ShowCreation(arrow2),
)
self.wait()
brace = Brace(VGroup(ps1.label, ps2.label), UP)
b_label = brace.get_text("5 bits").set_color(YELLOW)
self.play(
GrowFromCenter(brace),
FadeIn(b_label, 0.2 * UP),
)
self.wait()
def get_post_space(self, pre_space, n_bits):
n_rows = 2**((n_bits // 2))
n_cols = 2**((n_bits // 2) + n_bits % 2)
result = pre_space.get_grid(n_rows, n_cols, buff=0)
result.replace(pre_space, stretch=True)
result[:-1].set_fill(opacity=0)
return result
def get_post_space_labels(self, post_space, **kw):
n_bits = int(math.log2(len(post_space)))
top_label = OldTexText("Information = ", f"${n_bits}$ bits", **kw)
if n_bits == 1:
top_label[-1][-1].set_opacity(0)
top_label.next_to(post_space, UP, buff=0.15)
top_label.set_color(YELLOW)
bottom_label = OldTexText(f"$p = {{1 \\over {2**n_bits}}}$", **kw)
bottom_label.next_to(post_space, DOWN, SMALL_BUFF)
return VGroup(top_label, bottom_label)
def get_mini_group(self, pre_space, arrow, post_space, post_labels):
mini_group = VGroup(pre_space, arrow, post_space, post_labels[0]).copy()
mini_group.generate_target()
mini_group.target.scale(0.25)
mini_group.target[-1][0].set_opacity(0)
mini_group.target[-1][1].scale(3, about_edge=DOWN)
mini_group.target[-1][1].match_x(mini_group.target[2])
mini_group.target.next_to(post_space, RIGHT, buff=2.0)
mini_group[::2].set_fill(opacity=0)
mini_group.target[::2].set_stroke(width=1)
return mini_group
class AskForFormulaForI(Scene):
def construct(self):
tex = OldTex(
"I = ???",
tex_to_color_map={"I": YELLOW},
font_size=72,
)
tex.to_edge(UP)
self.play(Write(tex))
self.wait()
class MinusLogExpression(Scene):
def construct(self):
tex = OldTex(
"I = -\\log_2(p)",
tex_to_color_map={"I": YELLOW},
font_size=60,
)
self.play(FadeIn(tex, DOWN))
self.wait()
class ShowPatternInformationExamples(WordleDistributions):
def construct(self):
grid = self.grid
guess = "wordy"
self.add_word(guess)
axes = self.get_axes()
bars = self.get_distribution_bars(axes, guess)
index = 20
bar_indicator, x_tracker, get_bar = self.get_bar_indicator(bars, index)
p_label = self.get_p_label(get_bar)
I_label = self.get_information_label(p_label, get_bar)
I_label.scale(1.75)
I_label.add_updater(lambda m: m.next_to(grid, UR, buff=LARGE_BUFF).shift(0.5 * DOWN))
self.add(I_label)
randy = Randolph(height=2.0)
randy.flip()
randy.to_corner(DR)
self.play(PiCreatureSays(
randy, OldTexText("I thought this\\\\was a word game"),
bubble_config={"width": 4, "height": 3},
target_mode="pleading"
))
x_tracker.clear_updaters()
x_tracker.set_value(0)
self.play(
x_tracker.animate.set_value(120),
run_time=20,
rate_func=linear,
)
self.wait()
class TwentyBitOverlay(Scene):
def construct(self):
eq = OldTex("20 \\text{ bits} \\Leftrightarrow 0.00000095")
eq.scale(1.5)
self.play(Write(eq))
self.wait()
class AddingBitsObservationOverlay(Scene):
def construct(self):
mystery = Square(side_length=0.5).get_grid(1, 5, buff=SMALL_BUFF)
mystery.set_stroke(WHITE, 1)
for box in mystery:
char = Text("?", font="Consolas")
char.set_height(0.7 * box.get_height())
char.move_to(box)
box.add(char)
# Delete...
phase1_strs = [list("?????") for x in range(5)]
phase2_strs = [list("s????") for x in range(4)]
for i, s in enumerate(phase1_strs):
s[i] = "t"
for i, s in enumerate(phase2_strs):
s[i + 1] = "t"
kw = dict(
font="Consolas",
t2c={"t": RED, "s": YELLOW},
)
phase1 = VGroup(*(Text("".join(s), **kw) for s in phase1_strs))
phase2 = VGroup(*(Text("".join(s), **kw) for s in phase2_strs))
for phase in [phase1, phase2]:
phase.arrange(DOWN, buff=SMALL_BUFF)
phases = VGroup(mystery, phase1, phase2)
phases.arrange(RIGHT, buff=2.0)
phases.to_edge(UP)
# Arrange observations
mystery.set_x(-4).to_edge(UP, buff=LARGE_BUFF)
arrows = Arrow(LEFT, RIGHT).set_width(3).replicate(2)
arrows.arrange(RIGHT, buff=MED_LARGE_BUFF)
arrows.next_to(mystery, RIGHT, buff=MED_LARGE_BUFF)
obss = VGroup(
OldTexText("Has a `t'"),
OldTexText("Starts with `s'"),
)
obss.scale(0.85)
bits_labels = VGroup(
Text("2 bits"),
Text("3 more bits"),
)
bits_labels.scale(0.75)
for obs, bl, arrow in zip(obss, bits_labels, arrows):
obs.next_to(arrow, UP)
bl.next_to(arrow, DOWN)
bl.set_color(YELLOW)
self.add(mystery)
for arrow, bl, obs in zip(arrows, bits_labels, obss):
self.play(
ShowCreation(arrow),
Write(obs),
run_time=1
)
self.play(FadeIn(bl, 0.25 * DOWN))
self.wait()
long_arrow = Arrow(arrows[0].get_start(), arrows[1].get_end(), buff=0)
full_bits_label = Text("5 bits")
full_bits_label.match_style(bits_labels[0])
full_bits_label.next_to(long_arrow, DOWN)
self.play(bits_labels.animate.to_edge(UP, buff=SMALL_BUFF))
self.play(
FadeTransform(arrows[0], long_arrow),
FadeTransform(arrows[1], long_arrow),
Write(full_bits_label, run_time=1),
)
self.wait()
class ExpectedInformationLabel(Scene):
def construct(self):
eq = OldTex(
"E[\\text{Information}] = ",
"\\sum_x p(x) \\cdot \\text{Information}(x)",
tex_to_color_map={
"\\text{Information}": YELLOW,
},
font_size=60
)
eq.to_edge(UP)
self.play(Write(eq))
self.wait()
class LookTwoAheadWrapper(VideoWrapper):
title = "Later..."
class AskAboutPhysicsRelation(TeacherStudentsScene):
def construct(self):
physics = self.get_physics_entropy_image()
physics.next_to(self.students[2], UL)
physics.to_edge(UP)
self.student_says(
OldTexText("What does this have\\\\to do with thermodynamics?"),
target_mode="raise_right_hand",
index=2,
bubble_config=dict(width=5, height=3, direction=LEFT),
)
self.play(self.teacher.change("tease"))
self.play(
self.students[0].change("pondering", physics),
self.students[1].change("pondering", physics),
self.students[2].change("raise_left_hand", physics),
Write(physics),
)
self.wait(3)
self.embed()
def get_physics_entropy_image(self):
dots = Dot().get_grid(14, 10)
dots.set_height(3)
n = len(dots)
dots[:n // 2].set_color(RED)
dots[n // 2:].set_color(BLUE)
dots_copy = dots.deepcopy()
dots_copy.set_color(RED)
VGroup(*random.sample(list(dots_copy), n // 2)).set_color(BLUE)
pair = VGroup(dots, dots_copy)
pair.arrange(RIGHT, buff=LARGE_BUFF)
rects = VGroup(*map(SurroundingRectangle, pair))
rects.set_stroke(WHITE, 1)
labels = VGroup(
Text("Low entropy"),
Text("High entropy"),
)
labels.scale(0.75)
for label, rect in zip(labels, rects):
label.next_to(rect, UP)
return VGroup(labels, rects, pair)
class ContrastWearyAndSlate(WordleScene):
def construct(self):
grid = self.grid
grid.set_height(4)
grid.move_to(FRAME_WIDTH * LEFT / 4)
self.add_word("weary", wait_time_per_letter=0)
grid1 = grid.deepcopy()
self.add(grid1)
for x in range(5):
self.delete_letter()
grid.move_to(FRAME_WIDTH * RIGHT / 4)
self.add_word("slate", wait_time_per_letter=0)
grid2 = grid
grids = VGroup(grid1, grid2)
grids.to_edge(DOWN)
# Entropy
EI_label = OldTex(
"E[I]", "= ", "\\sum_{x} p(x) \\log_2\\big(1 / p(x) \\big)",
tex_to_color_map={"I": BLUE},
font_size=36,
)
EI_label.to_edge(UP)
EI_rect = SurroundingRectangle(EI_label)
EI_rect.set_stroke(YELLOW, 2)
values = VGroup(
DecimalNumber(4.90, unit="\\text{ bits}"),
DecimalNumber(5.87, unit="\\text{ bits}"),
)
arrows = VGroup()
for value, grid in zip(values, grids):
value[4:].shift(SMALL_BUFF * RIGHT)
value.next_to(grid, UP)
arrows.add(Arrow(EI_rect, value))
self.add(EI_label)
self.add(EI_rect)
for arrow, value in zip(arrows, values):
self.play(
ShowCreation(arrow),
CountInFrom(value)
)
self.wait()
class VonNeumannPhrase(Scene):
text = "You should call \n it entropy!"
def construct(self):
label = Text(self.text)
label.set_backstroke(width=8)
for word in self.text.split():
self.add(label.get_part_by_text(word))
self.wait(0.1)
self.wait()
class VonNeumannPhrase2(VonNeumannPhrase):
text = "Nobody knows what \n entropy really is."
class MaximumInsert(Scene):
def construct(self):
text = OldTexText("Maximum possible\\\\", "expected information")
arrow = Arrow(text.get_top(), text.get_top() + UR)
arrow.shift(RIGHT)
VGroup(text, arrow).set_color(YELLOW)
self.play(
Write(text),
ShowCreation(arrow),
run_time=1
)
self.wait()
class ShowEntropyCalculations(IntroduceDistribution):
grid_height = 3.5
grid_center = [-5.0, -1.0, 0]
CONFIG = {"random_seed": 0}
n_words = 100
def construct(self):
# Axes
grid = self.grid
kw = dict(x_max=150, width=8.5, height=6.5)
axes = self.get_axes(y_max=0.2, **kw)
axes.to_edge(RIGHT, buff=0.1)
axes.to_edge(UP, buff=0.5)
y_label = OldTex("p", font_size=24)
y_label.next_to(axes.y_axis.n2p(0.2), RIGHT)
axes.y_axis.add(y_label)
self.add(axes)
# old_axes = self.get_axes(y_max=0.4, **kw)
# old_axes.next_to(axes, DOWN, buff=0.8)
# y_label = OldTex("p \\cdot \\log_2(1/p)", font_size=24)
# y_label.next_to(old_axes.y_axis.n2p(0.4), RIGHT, MED_SMALL_BUFF)
# old_axes.y_axis.add(y_label)
# self.add(old_axes)
# Formula
ent_formula = self.get_entropy_label()
ent_formula.scale(1.2)
ent_formula.move_to(axes, UR)
ent_formula.shift(DOWN)
ent_formula.shift_onto_screen()
ent_rhs = ent_formula[1]
self.add(ent_formula)
n = 3**5
# Bang on through
words = list(random.sample(self.all_words, self.n_words))
words = ["maths", "weary", "other", "tares", "kayak"] + words
for word in words:
low_bars = self.get_distribution_bars(axes, word)
self.add(low_bars)
dist = np.array([bar.prob for bar in low_bars])
ent_summands = -np.log2(dist, where=dist > 1e-10) * dist
# high_bars = self.get_bars(old_axes, ent_summands)
# high_bars.add_updater(lambda m: m.match_style(low_bars))
# self.add(high_bars)
self.add_word(word, wait_time_per_letter=0)
trackers = self.add_trackers(low_bars, index=0)
x_tracker, bar_indicator, p_label, info_label, match_label = trackers
p_label.add_updater(lambda m: m.move_to(axes, DL).shift([2, 1, 0]))
self.remove(info_label)
self.remove(match_label)
# Highlight answer
arrow = OldTex("\\rightarrow")
pw = grid.pending_word.copy()
pw.generate_target()
pw.arrange(RIGHT, buff=0.05)
rhs = ent_rhs.copy()
rhs.set_value(sum(ent_summands))
group = VGroup(pw, arrow, rhs)
group.set_color(BLUE)
group.arrange(RIGHT)
group.match_width(grid)
group.next_to(grid, UP, LARGE_BUFF)
self.add(group)
# Show calculation
x_tracker.suspend_updating()
n = list(dist).index(0) + 1
self.play(
UpdateFromAlphaFunc(
x_tracker, lambda m, a: m.set_value(int(a * (n - 1))),
),
UpdateFromAlphaFunc(
ent_rhs, lambda m, a: m.set_value(sum(ent_summands[:int(a * n)]))
),
rate_func=linear,
run_time=4 * n / 3**5,
)
self.wait()
# x_tracker.resume_updating()
# self.embed()
self.remove(group)
# Clear
self.remove(*trackers, low_bars, pw, arrow, rhs)
ent_rhs.set_value(0)
grid.pending_word.set_submobjects([])
grid.clear_updaters()
grid.add_updater(lambda m: m)
self.get_curr_row().set_fill(BLACK, 0)
class WrapperForEntropyCalculation(VideoWrapper):
title = "Search for maximum entropy"
class AltWrapperForEntropyCalculation(VideoWrapper):
title = "Refined entropy calculation"
animate_boundary = False
class UniformPriorExample(WordleSceneWithAnalysis):
uniform_prior = True
show_prior = False
weight_to_prob = 0 # TODO
pre_computed_first_guesses = [
"tares", "lares", "rales", "rates", "teras",
"nares", "soare", "tales", "reais", "tears",
"arles", "tores", "salet",
]
class MentionUsingWordFrequencies(TeacherStudentsScene):
def construct(self):
self.teacher_says(
OldTexText("Next step: Integrate\\\\word frequency data"),
added_anims=[self.change_students("hooray", "happy", "tease")]
)
self.wait()
self.play(self.students[1].change("pondering"))
self.wait(2)
class V2TitleCard(TitleCardScene):
n_letters = 6
words = ["how-to", "prefer", "common", "words"]
secret_word = "priors"
class HowThePriorWorks(Scene):
def construct(self):
# Prepare columns
all_words = get_word_list()
freq_map = get_word_frequencies()
sorted_words = list(sorted(all_words, key=lambda w: -freq_map[w]))
col1, col2 = cols = [
WordleScene.get_grid_of_words(
word_list, 25, 1, dots_index=-12
)
for word_list in (random.sample(all_words, 100), sorted_words)
]
for col in cols:
col.set_height(6)
col.set_x(-1)
col.to_edge(DOWN, buff=MED_SMALL_BUFF)
bars1, bars2 = [
self.get_freq_bars(col, freq_map, max_width=width, exp=exp)
for col, width, exp in zip(cols, (1, 2), (0.3, 1))
]
group1 = VGroup(col1, bars1)
group2 = VGroup(col2, bars2)
col1_title = VGroup(
Text("Relative frequencies of all words"),
Text("From the Google Books English n-gram public dataset", font_size=24, color=GREY_B),
)
col1_title.arrange(DOWN)
col1_title.set_height(1)
col1_title.next_to(col1, UP, MED_LARGE_BUFF)
# Introduce frequencies
for bar in bars1:
bar.save_state()
bar.stretch(0, 0, about_edge=LEFT)
bar.set_stroke(width=0)
self.wait()
self.add(col1)
self.play(
LaggedStartMap(Restore, bars1),
FadeIn(col1_title, 0.5 * UP)
)
self.wait()
arrow = Vector(2 * RIGHT, stroke_width=5)
arrow.set_x(0).match_y(col1)
arrow_label = Text("Sort", font_size=36)
arrow_label.next_to(arrow, UP, SMALL_BUFF)
self.play(
ShowCreation(arrow),
Write(arrow_label),
group1.animate.next_to(arrow, LEFT)
)
group2.next_to(arrow, RIGHT, buff=LARGE_BUFF)
self.play(LaggedStart(*(
FadeInFromPoint(VGroup(word, bar), col1.get_center())
for word, bar in zip(col2, bars2)
), lag_ratio=0.1, run_time=3))
self.wait()
# Word play
numbers = VGroup(
*(Integer(i + 1) for i in range(13))
)
numbers.match_height(col2[0])
for number, word in zip(numbers, col2):
number.next_to(word, LEFT, SMALL_BUFF, aligned_edge=UP)
number.word = word
number.add_updater(lambda m: m.match_style(m.word))
rect = SurroundingRectangle(col2[:13], buff=0.05)
rect.set_stroke(YELLOW, 2)
self.play(ShowCreation(rect))
self.wait()
self.play(
rect.animate.replace(col2[7], stretch=True).set_opacity(0),
col2[7].animate.set_color(YELLOW),
ShowIncreasingSubsets(numbers),
run_time=0.5
)
self.wait()
self.remove(rect)
for i in [0, 1, 2, 8, 7, 6, 3, 4, (9, 10, 11, 12)]:
col2.set_color(GREY_A)
for j in listify(i):
col2[j].set_color(YELLOW)
self.wait(0.5)
self.play(col2.animate.set_color(GREY_A))
# Don't care about relative frequencies
comp_words = ["which", "braid"]
which_group, braid_group = comp = VGroup(*(
VGroup(
Text(word, font="Consolas"),
Vector(RIGHT),
DecimalNumber(freq_map[word], num_decimal_places=6)
).arrange(RIGHT)
for word in comp_words
))
comp.arrange(DOWN, buff=2.0)
comp.to_edge(LEFT)
percentages = DecimalNumber(99.9, num_decimal_places=1, unit="\\%").replicate(2)
rhss = VGroup()
for per, group in zip(percentages, comp):
rhs = group[2]
rhss.add(rhs)
per.move_to(rhs, LEFT)
rhs.generate_target()
rhs.target.scale(0.8)
rhs.target.set_color(GREY_B)
rhs.target.next_to(per, DOWN, aligned_edge=LEFT)
self.play(
FadeOut(arrow),
FadeOut(arrow_label),
FadeOut(group1),
FadeTransform(col2[0].copy(), which_group[0]),
)
self.play(
ShowCreation(which_group[1]),
CountInFrom(which_group[2], 0),
)
self.wait()
self.play(FadeTransform(which_group[:2].copy(), braid_group[:2]))
self.play(CountInFrom(braid_group[2], 0, run_time=0.5))
self.wait()
self.play(
FadeIn(percentages, 0.75 * DOWN),
*map(MoveToTarget, rhss),
)
self.wait()
# Sigmoid
axes = Axes((-10, 10), (0, 2, 0.25), width=12, height=6)
axes.y_axis.add_numbers(np.arange(0.25, 2.25, 0.25), num_decimal_places=2, font_size=18)
axes.center()
col3 = WordleScene.get_grid_of_words(sorted_words, 25, 4, dots_index=-50)
col3.arrange(DOWN, buff=SMALL_BUFF)
col3.generate_target()
col3.target.rotate(-90 * DEGREES)
col3.target.match_width(axes.x_axis)
col3.target.next_to(axes.x_axis, DOWN, buff=0)
col2_words = [w.text for w in col2]
col3.match_width(col2)
for word in col3:
if word.text in col2_words:
word.move_to(col2[col2_words.index(word.text)])
else:
word.rotate(-90 * DEGREES)
word.move_to(col2[col2_words.index('...')])
word.scale(0)
word.set_opacity(0)
self.remove(col2),
self.play(LaggedStart(
FadeOut(VGroup(comp, percentages), 2 * LEFT),
FadeOut(numbers),
FadeOut(bars2),
FadeOut(col1_title, UP),
MoveToTarget(col3),
Write(axes),
FadeOut(col2[col2_words.index("...")]),
run_time=5,
))
self.wait()
graph = axes.get_graph(sigmoid)
graph.set_stroke(BLUE, 3)
graph_label = OldTex("\\sigma(x) = {1 \\over 1 + e^{-x} }")
graph_label.next_to(graph.get_end(), UL)
self.play(ShowCreation(graph))
self.play(Write(graph_label))
self.wait()
# Lines to graph
lines = Line().replicate(len(col3))
lines.set_stroke(BLUE_B, 1.0)
def update_lines(lines):
for line, word in zip(lines, col3):
line.put_start_and_end_on(
word.get_top(),
axes.input_to_graph_point(axes.x_axis.p2n(word.get_center()), graph),
)
lines.add_updater(update_lines)
self.play(ShowCreation(lines, lag_ratio=0.05, run_time=5))
self.wait()
self.play(col3.animate.scale(2.0, about_edge=UP), run_time=3)
self.play(col3.animate.scale(0.25, about_edge=UP), run_time=3)
self.play(col3.animate.scale(2.0, about_edge=UP), run_time=3)
self.wait()
for vect in [RIGHT, 2 * LEFT, RIGHT]:
self.play(col3.animate.shift(vect), run_time=2)
lines.clear_updaters()
self.wait()
# Show window of words
n_shown = 15
col4 = WordleScene.get_grid_of_words(
sorted_words[3000:3000 + n_shown], 20, 1
)
dots = Text("...", font="Consolas", font_size=24).rotate(90 * DEGREES)
col4.add_to_back(dots.copy().next_to(col4, UP))
col4.add(dots.copy().next_to(col4, DOWN))
col4.set_height(6)
col4.to_corner(UL)
col4.shift(RIGHT)
numbers = VGroup(*(Integer(n) for n in range(3000, 3000 + n_shown)))
numbers.set_height(col4[1].get_height())
for number, word in zip(numbers, col4[1:]):
number.next_to(word, LEFT, MED_SMALL_BUFF, aligned_edge=UP)
number.match_style(word)
number.align_to(numbers[0], LEFT)
word.add(number)
self.play(ShowIncreasingSubsets(col4))
self.wait()
def get_freq_bars(self, words, freq_map, max_width=2, exp=1):
freqs = [freq_map.get(w.text, 0)**exp for w in words] # Smoothed out a bit
max_freq = max(freqs)
bars = VGroup()
height = np.mean([w.get_height() for w in words]) * 0.8
for word, freq in zip(words, freqs):
bar = Rectangle(
height=height,
width=max_width * freq / max_freq,
stroke_color=WHITE,
stroke_width=1,
fill_color=BLUE,
fill_opacity=1,
)
bar.next_to(word, RIGHT, SMALL_BUFF)
if word.text not in freq_map:
bar.set_opacity(0)
bars.add(bar)
return bars
class ShowWordLikelihoods(Scene):
title = "How likely is each word\\\\to be an answer?"
n_shown = 20
def construct(self):
all_words = get_word_list()
n = self.n_shown
words = random.sample(all_words, n)
word_mobs = WordleScene.get_grid_of_words(words, 2, n // 2)
word_mobs[n // 2:].next_to(word_mobs[:n // 2], DOWN, buff=2.0)
word_mobs.set_width(FRAME_WIDTH - 2)
word_mobs.to_edge(DOWN)
self.add(word_mobs)
title = OldTexText(self.title)
title.to_edge(UP)
self.add(title)
freq_prior = get_frequency_based_priors()
true_prior = get_true_wordle_prior()
bars = VGroup()
decs = VGroup()
for word in word_mobs:
fp = freq_prior[word.text]
tp = true_prior[word.text]
p = (0.7 * tp + 0.3 * fp)
bar = Rectangle(0.5, 1.5 * p)
bar.set_stroke(WHITE, 1)
bar.set_fill(BLUE, 1)
bar.next_to(word, UP, SMALL_BUFF)
dec = DecimalNumber(100 * p, unit="\\%")
dec.scale(0.5)
dec.bar = bar
dec.add_updater(lambda m: m.next_to(m.bar, UP, SMALL_BUFF))
dec.next_to(bar, UP)
bar.save_state()
bar.stretch(0, 1, about_edge=DOWN)
bar.set_opacity(0)
decs.add(dec)
bars.add(bar)
self.play(
LaggedStartMap(Restore, bars, lag_ratio=0.01),
LaggedStartMap(CountInFrom, decs, lag_ratio=0.01),
run_time=4
)
self.wait()
self.bars = bars
self.decs = decs
self.word_mobs = word_mobs
class SidewaysWordProbabilities(Scene):
CONFIG = {"random_seed": 5}
def construct(self):
# Blatant copy-paste-and-modify from scene above...
all_words = get_word_list()
n = 15
words = random.sample(all_words, n)
word_mobs = WordleScene.get_grid_of_words(words, 15, 1)
word_mobs.arrange(DOWN, buff=MED_SMALL_BUFF)
word_mobs.set_height(FRAME_HEIGHT - 1)
word_mobs.to_edge(LEFT)
self.add(word_mobs)
freq_prior = get_frequency_based_priors()
true_prior = get_true_wordle_prior()
bars = VGroup()
decs = VGroup()
for word in word_mobs:
fp = freq_prior[word.text]
tp = true_prior[word.text]
p = (0.7 * tp + 0.3 * fp)
bar = Rectangle(1.5 * p, 0.2)
bar.set_stroke(WHITE, 1)
bar.set_fill(BLUE, 1)
bar.next_to(word, RIGHT, MED_SMALL_BUFF)
dec = DecimalNumber(100 * p, unit="\\%", num_decimal_places=0)
dec.scale(0.5)
dec.bar = bar
dec.add_updater(lambda m: m.next_to(m.bar, RIGHT, SMALL_BUFF))
bar.save_state()
bar.stretch(0, 0, about_edge=LEFT)
bar.set_opacity(0)
decs.add(dec)
bars.add(bar)
self.play(
LaggedStartMap(Restore, bars, lag_ratio=0.01),
LaggedStartMap(CountInFrom, decs, lag_ratio=0.01),
run_time=4
)
self.wait()
class DistributionOverWord(ShowWordLikelihoods):
CONFIG = {"random_seed": 2}
class LookThroughWindowsOfWords(Scene):
def construct(self):
# Copied from previous scene
priors = get_frequency_based_priors()
sorted_words = sorted(get_word_list(), key=lambda w: -priors[w])
base = 2880
group_size = 20
n_groups = 3
col4 = VGroup(*(
WordleScene.get_grid_of_words(
sorted_words[base + n * group_size:base + (n + 1) * group_size],
group_size, 1
)
for n in range(n_groups)
))
col4.arrange(DOWN, buff=0.1)
col4 = VGroup(*it.chain(*col4))
col4.center().to_edge(UP)
numbers = VGroup(*(Integer(n) for n in range(base, base + n_groups * group_size)))
numbers.set_height(col4[1].get_height())
for number, word in zip(numbers, col4[1:]):
number.next_to(word, LEFT, MED_SMALL_BUFF, aligned_edge=UP)
number.match_style(word)
number.align_to(numbers[0], LEFT)
number.align_to(word[np.argmin([c.get_height() for c in word])], DOWN)
word.add(number)
self.add(col4)
frame = self.camera.frame
self.play(frame.animate.align_to(col4, DOWN), run_time=20)
class EntropyOfWordDistributionExample(WordleScene):
grid_height = 4
grid_center = 4.5 * LEFT
secret_word = "graph"
wordle_based_prior = True
def construct(self):
# Try first two guesses
grid = self.grid
guesses = ["other", "nails"]
if not self.presenter_mode or self.skip_animations:
self.add_word("other")
self.reveal_pattern()
self.add_word("nails")
self.reveal_pattern()
else:
self.wait()
self.wait("Enter \"{}\" then \"{}\"".format(*guesses))
# Add match label
match_label = VGroup(Integer(4, edge_to_fix=RIGHT), Text("Matches"))
match_label.scale(0.75)
match_label.arrange(RIGHT, buff=MED_SMALL_BUFF)
match_label.next_to(grid, UP)
self.add(match_label)
# Show words
s_words = get_word_list(short=True)
col1 = self.get_grid_of_words(
sorted(list(set(self.possibilities).intersection(s_words))),
4, 1
)
col1.scale(1.5)
col1.next_to(grid, RIGHT, buff=1)
bars1 = VGroup(*(
self.get_prob_bar(word, 0.25)
for word in col1
))
for bar in bars1:
bar.save_state()
bar.stretch(0, 0, about_edge=LEFT)
bar.set_opacity(0)
self.play(
ShowIncreasingSubsets(col1),
CountInFrom(match_label[0], 0),
)
self.wait()
self.play(LaggedStartMap(Restore, bars1))
self.wait()
# Ask about entropy
brace = Brace(bars1, RIGHT)
question = Text("What is the\nentropy?", font_size=36)
question.next_to(brace, RIGHT)
formula = OldTex(
"H &=",
"\\sum_x p(x) \\cdot", "\\log_2\\big(1 / p(x) \\big)\\\\",
font_size=36,
)
formula.next_to(brace, RIGHT, submobject_to_align=formula[0])
info_box = SurroundingRectangle(formula[2], buff=SMALL_BUFF)
info_box.set_stroke(TEAL, 2)
info_label = Text("Information", font_size=36)
info_label.next_to(info_box, UP)
info_label.match_color(info_box)
info_value = OldTex("\\log_2(4)", "=", "2", font_size=36)
info_value[1].rotate(PI / 2)
info_value.arrange(DOWN, SMALL_BUFF)
info_value.next_to(info_box, DOWN)
alt_lhs = formula[0].copy().next_to(info_value[-1], LEFT)
self.play(
GrowFromCenter(brace),
Write(question),
)
self.wait()
self.play(
FadeIn(formula, lag_ratio=0.1),
question.animate.shift(2 * UP)
)
self.wait()
self.play(
ShowCreation(info_box),
Write(info_label)
)
self.wait()
self.play(FadeIn(info_value[0]))
self.wait()
self.play(Write(info_value[1:]))
self.wait()
self.play(TransformFromCopy(formula[0], alt_lhs))
self.wait()
# Introduce remaining words
col2 = self.get_grid_of_words(
sorted(self.possibilities), 16, 1
)
col2.match_width(col1)
col2.move_to(col1, LEFT)
col2.save_state()
col1_words = [w.text for w in col1]
for word in col2:
if word.text in col1_words:
word.move_to(col1[col1_words.index(word.text)])
else:
word.move_to(col1)
word.set_opacity(0)
pre_bars2, bars2 = [
VGroup(*(
self.get_prob_bar(
word,
0.246 * self.priors[word.text] + 0.001,
num_decimal_places=3,
)
for word in group
))
for group in (col2, col2.saved_state)
]
new_brace = Brace(bars2, RIGHT)
self.play(
FadeTransform(col1, col2),
FadeTransform(bars1, pre_bars2),
LaggedStart(*map(FadeOut, [alt_lhs, info_value, info_label, info_box]))
)
self.play(
ChangeDecimalToValue(match_label[0], 16, run_time=1),
Restore(col2, run_time=2),
ReplacementTransform(pre_bars2, bars2, run_time=2),
Transform(brace, new_brace),
)
self.wait()
# Pass the time by entering the various words
shuffled_col2 = list(col2)
random.shuffle(shuffled_col2)
for word in shuffled_col2:
if word.text in [w.text for w in col1]:
continue
word.set_color(YELLOW)
for letter in word.text:
self.add_letter(letter)
self.wait(random.random() * 0.2)
self.wait(0.5)
for x in range(5):
self.delete_letter()
self.wait(0.1)
word.set_color(WHITE)
shuffled_col2 = list(col2)
random.shuffle(shuffled_col2)
for word in shuffled_col2:
if word.text not in [w.text for w in col1]:
continue
rect = SurroundingRectangle(word)
rect.set_color(GREEN)
self.add(rect)
word.set_color(GREEN)
for letter in word.text:
self.add_letter(letter)
self.wait(random.random() * 0.2)
self.wait(0.5)
for x in range(5):
self.delete_letter()
self.wait(0.1)
word.set_color(WHITE)
self.remove(rect)
# Proposed answer
rhs1 = OldTex("= \\log_2(16) = 4?", font_size=36)
rhs1.next_to(formula[1], DOWN, aligned_edge=LEFT)
cross = Cross(rhs1).set_stroke(RED, 6)
rhs2 = OldTex(
"= &4 \\big(0.247 \\cdot \\log_2(1/0.247)\\big) \\\\",
"+ &12 \\big(0.001 \\cdot \\log_2(1/0.001)\\big)\\\\ ",
"= &2.11",
font_size=30,
)
rhs2.next_to(rhs1, DOWN, aligned_edge=LEFT)
self.play(Write(rhs1))
self.wait()
self.play(ShowCreation(cross))
self.wait()
self.play(
Write(rhs2),
run_time=3
)
self.wait()
rect = SurroundingRectangle(rhs2[-1])
self.play(ShowCreation(rect))
self.wait()
def get_prob_bar(self, word, prob, num_decimal_places=2, height=0.15, width_mult=8.0):
bar = Rectangle(
height=height,
width=width_mult * prob,
stroke_color=WHITE,
stroke_width=1,
fill_color=BLUE,
)
bar.next_to(word, RIGHT, MED_SMALL_BUFF)
label = DecimalNumber(prob, font_size=24, num_decimal_places=num_decimal_places)
label.next_to(bar, RIGHT, SMALL_BUFF)
bar.add(label)
bar.label = label
bar.set_opacity(word[0].get_fill_opacity())
return bar
def seek_good_examples(self):
words = get_word_list()
swords = get_word_list(short=True)
for answer in swords:
poss = list(words)
for guess in ["other", "nails"]:
poss = get_possible_words(
guess,
get_pattern(guess, answer),
poss,
)
n = len(set(poss).intersection(swords))
m = len(poss)
if n == 4 and m in (16, 32, 64):
print(answer, n, len(poss))
class WhatMakesWordleNice(TeacherStudentsScene):
def construct(self):
self.teacher_says(
OldTexText("This is what makes wordle\\\\such a nice example"),
added_anims=[self.change_students(
"pondering", "thinking", "erm",
look_at=ORIGIN,
)]
)
self.wait(5)
class TwoInterpretationsWrapper(Scene):
def construct(self):
self.add(FullScreenRectangle())
screens = ScreenRectangle().get_grid(1, 2, buff=MED_LARGE_BUFF)
screens.set_fill(BLACK, 1)
screens.set_stroke(WHITE, 1)
screens.set_width(FRAME_WIDTH - 1)
screens.move_to(DOWN)
title = Text("Two applications of entropy", font_size=60)
title.to_edge(UP)
screen_titles = VGroup(
Text("Expected information from guess"),
Text("Remaining uncertainty"),
)
screen_titles.scale(0.8)
for screen, word in zip(screens, screen_titles):
word.next_to(screen, UP)
screen_titles[0].set_color(BLUE)
screen_titles[1].set_color(TEAL)
self.add(title)
self.add(screens)
self.wait()
for word in screen_titles:
self.play(Write(word, run_time=1))
self.wait()
class IntroduceDistributionFreqPrior(IntroduceDistribution):
n_word_rows = 1
uniform_prior = False
show_fraction_in_p_label = False
class FreqPriorExample(WordleSceneWithAnalysis):
pre_computed_first_guesses = [
"tares", "lares", "rates", "rales", "tears",
"tales", "salet", "teras", "arles", "nares",
"soare", "saner", "reals"
]
class ConstrastResultsWrapper(Scene):
def construct(self):
self.add(FullScreenRectangle())
screens = Rectangle(4, 3).replicate(2)
screens.arrange(RIGHT, buff=SMALL_BUFF)
screens.set_width(FRAME_WIDTH - 1)
screens.set_stroke(WHITE, 1)
screens.set_fill(BLACK, 1)
screens.move_to(DOWN)
self.add(screens)
class WordlePriorExample(WordleSceneWithAnalysis):
secret_word = "thump"
wordle_based_prior = True
pre_computed_first_guesses = [
"soare", "raise", "roate", "raile", "reast",
"slate", "crate", "irate", "trace", "salet",
"arise", "orate", "stare"
]
class HowToCombineEntropyAndProbability(FreqPriorExample):
secret_word = "words"
def construct(self):
super().construct()
# Put in first three
guesses = ["favor", "ideal", "scores"]
if not self.presenter_mode or self.skip_animations:
for guess in guesses:
self.add_word(guess)
self.reveal_pattern()
else:
self.wait(note=f"Enter{guesses}, discuss")
# Fade lower grid
rows = self.guess_value_grid
self.wait()
self.play(rows[:2].animate.set_opacity(0.3))
self.wait()
self.play(
rows[:2].animate.set_opacity(1),
rows[2:].animate.set_opacity(0.3),
)
self.wait()
self.play(
rows[1].animate.set_opacity(0.3),
rows[2].animate.set_opacity(1),
rows[3:].animate.set_opacity(0.3),
)
self.wait()
# Expected score
es_eq = OldTex(
"E[\\text{Score}] = 0.58 \\cdot {4} +",
"(1 - 0.58)", " \\cdot \\big({4} + f(1.44 - 1.27)\\big)",
tex_to_color_map={
"\\text{Score}": YELLOW,
"{4}": YELLOW,
"0.58": self.prior_color,
"1.44": self.entropy_color,
"1.27": self.entropy_color,
"=": WHITE,
}
)
es_eq.next_to(self.grid, DOWN, LARGE_BUFF)
es_eq.to_edge(RIGHT)
left_part = es_eq[:11]
left_part.save_state()
left_part.align_to(self.grid, LEFT)
q_marks = OldTex("???")
q_marks.next_to(es_eq.get_part_by_tex("="), RIGHT)
if not self.presenter_mode or self.skip_animations:
self.add_word("words")
else:
self.wait(note="Enter \"words\"")
self.wait()
self.play(
Write(es_eq[:4]),
FadeIn(q_marks)
)
self.wait()
self.play(FlashAround(rows[0][2], run_time=3))
self.play(
FadeTransform(rows[0][2].copy(), es_eq.get_part_by_tex("0.58")),
FadeIn(es_eq.slice_by_tex("\\cdot", "(1 -")),
q_marks.animate.next_to(es_eq[:8], RIGHT, aligned_edge=UP)
)
self.remove(es_eq)
self.add(es_eq[:8])
self.wait()
self.play(
FadeIn(es_eq[8:11]),
q_marks.animate.next_to(es_eq[10], RIGHT),
FadeOut(rows[3:])
)
self.wait()
self.play(
Restore(left_part),
FadeTransform(q_marks, es_eq[11:])
)
self.wait()
class FirstThoughtsOnCombination(Scene):
def construct(self):
morty = Mortimer(height=2)
morty.flip()
morty.to_corner(DL)
example = VGroup(
Text("dorms", font="Consolas"),
DecimalNumber(1.08, color=TEAL),
DecimalNumber(0.31, color=BLUE),
)
word, n1, n2 = example
example.arrange(RIGHT, buff=1.5)
example[0].shift(0.5 * RIGHT)
example.to_corner(UR)
example.shift(DOWN)
example_titles = VGroup(
OldTex("E[\\text{Info.}]", color=TEAL),
OldTex("p(\\text{word})", color=BLUE),
)
for title, ex in zip(example_titles, example[1:]):
title.next_to(ex, UP, MED_LARGE_BUFF)
title.add(Underline(title, buff=-0.05).set_stroke(GREY, 1))
self.add(example_titles)
self.add(example)
self.add(morty)
self.play(PiCreatureBubbleIntroduction(
morty,
OldTexText("How should I measure\\\\guess quality?", font_size=36),
look_at=example,
target_mode="pondering",
bubble_type=ThoughtBubble,
bubble_config={"width": 4, "height": 3},
))
self.play(Blink(morty))
self.wait()
attempt = example.copy()
attempt.generate_target()
arrow = OldTex("\\rightarrow")
plus = OldTex("+")
group = VGroup(
attempt.target[0],
arrow,
attempt.target[1],
plus,
attempt.target[2],
)
group.arrange(RIGHT, buff=0.2)
group.next_to(example, DOWN, buff=2)
self.play(
MoveToTarget(attempt),
morty.change("shruggie", attempt),
FadeIn(arrow), FadeIn(plus),
)
self.wait()
self.play(Blink(morty))
cross = Cross(group)
cross.insert_n_curves(100)
better_words = OldTexText("We can do\\\\better!")
better_words.next_to(cross, DOWN)
better_words.set_color(RED)
self.play(ShowCreation(cross))
self.play(
Write(better_words, run_time=1),
morty.change("pondering", cross),
)
self.play(Blink(morty))
self.wait()
self.embed()
class EntropyToScoreData(Scene):
def construct(self):
# Axes
axes = Axes(
(0, 13), (0, 6),
height=6,
width=10,
)
axes.x_axis.add_numbers()
axes.y_axis.add_numbers()
x_label = Text("Entropy", font_size=24)
x_label.next_to(axes.x_axis, UR, SMALL_BUFF)
x_label.shift_onto_screen()
y_label = Text("Score", font_size=24)
y_label.next_to(axes.y_axis, UR)
y_label.shift_onto_screen()
axes.add(x_label, y_label)
self.add(axes)
self.wait()
self.wait()
# Data
with open(ENT_SCORE_PAIRS_FILE) as fp:
data = np.array(json.load(fp))
dots = DotCloud([
axes.c2p(*pair)
for pair in data
])
dots.set_radius(0.05)
dots.set_color(BLUE)
dots.set_opacity(0.02)
dots.add_updater(lambda m: m)
self.play(ShowCreation(dots, run_time=3))
self.wait()
# Window
window = FullScreenRectangle().replicate(2)
window.arrange(RIGHT, buff=3 * dots.radius)
window.set_fill(BLACK, 0.7)
window.set_x(axes.c2p(8.65, 0)[0])
self.add(dots, window, axes)
self.wait()
for x in (0, 1, 2):
self.play(
window.animate.set_x(axes.c2p(x, 0)[0])
)
self.wait()
self.play(FadeOut(window))
self.wait()
# Buckets
bucket_size = 0.25
buckets = dict()
bucket_xs = np.arange(0, axes.x_range[1], bucket_size)
bars = VGroup()
for x in bucket_xs:
indices = (x < data[:, 0]) & (data[:, 0] <= x + bucket_size)
buckets[x] = data[indices, 1]
y = data[indices, 1].mean()
if not indices.any():
continue
bar = Rectangle(
width=axes.x_axis.unit_size * bucket_size,
height=axes.y_axis.unit_size * y
)
bar.set_stroke(WHITE, 1)
bar.move_to(axes.c2p(x, 0), DL)
bar.set_fill(TEAL, 0.5)
bars.add(bar)
self.play(FadeIn(bars, lag_ratio=0.1, run_time=2))
self.wait()
bars.save_state()
self.play(
bars[:4].animate.fade(0.7),
bars[5:].animate.fade(0.7),
)
self.wait()
self.play(Restore(bars))
self.play(
bars[:16].animate.fade(0.7),
bars[17:].animate.fade(0.7),
)
self.wait()
self.play(Restore(bars))
# Model
curve = axes.get_graph(
lambda x: 1 + 0.56 * math.log(x + 1) + 0.08 * x,
)
curve.set_stroke(YELLOW, 2)
self.play(ShowCreation(curve, run_time=2))
self.wait()
class LookTwoStepsAhead(WordleSceneWithAnalysis):
look_two_ahead = True
wordle_based_prior = True
pre_computed_first_guesses = [
"slate", "salet", "slane", "reast", "trace",
"carse", "crate", "torse", "carle", "carte",
"toile", "crane", "least", "saint", "crine",
"roast",
]
class HowLookTwoAheadWorks(Scene):
prob_color = BLUE_D
entropy_color = TEAL
first_guess = "tares"
n_shown_trials = 240
transition_time = 1
def get_priors(self):
return get_frequency_based_priors()
def construct(self):
# Setup
all_words = get_word_list()
possibilities = get_word_list()
priors = self.get_priors()
# Show first guess
guess1 = self.get_word_mob(self.first_guess)
guess1.to_edge(LEFT)
pattern_array1 = self.get_pattern_array(guess1, possibilities, priors)
prob_bars1 = self.get_prob_bars(pattern_array1.pattern_mobs)
self.add(guess1)
self.play(
ShowCreation(
pattern_array1.connecting_lines,
lag_ratio=0.1
),
LaggedStartMap(
FadeIn, pattern_array1.pattern_mobs,
shift=0.2 * RIGHT,
lag_ratio=0.1,
),
run_time=2,
)
self.play(Write(pattern_array1.dot_parts))
self.wait()
for bar in prob_bars1:
bar.save_state()
bar.stretch(0, 0, about_edge=LEFT)
bar.set_opacity(0)
self.play(LaggedStartMap(Restore, prob_bars1))
self.wait()
# Reminder on entropy
H_eq = OldTex(
"E[I] = \\sum_{x} p(x) \\cdot \\log_2\\big((1 / p(x)\\big)",
font_size=36
)
H_eq.next_to(prob_bars1, RIGHT)
info_labels = VGroup(*(
self.get_info_label(bar)
for bar in prob_bars1
))
self.play(Write(H_eq))
self.wait()
self.play(FadeIn(info_labels[0], lag_ratio=0.1))
self.wait()
self.play(
LaggedStartMap(FadeIn, info_labels[1:], lag_ratio=0.5),
H_eq.animate.scale(0.7).to_edge(DOWN),
run_time=2,
)
self.wait()
H_label = self.get_entropy_label(guess1, pattern_array1.distribution)
self.play(FadeTransform(H_eq, H_label))
self.wait()
self.play(LaggedStartMap(FadeOut, info_labels), run_time=1)
# Show example second guess
word_buckets = get_word_buckets(guess1.text, possibilities)
arrows = VGroup()
second_guesses = VGroup()
second_ents = VGroup()
for i, bar in enumerate(prob_bars1):
pattern = pattern_array1.pattern_mobs[i].pattern
bucket = word_buckets[pattern]
optimal_word = optimal_guess(all_words, bucket, priors)
shown_words = random.sample(all_words, self.n_shown_trials)
shown_words.append(optimal_word)
for j, shown_word in enumerate(shown_words):
guess2 = self.get_word_mob(shown_word)
guess2.set_width(0.7)
guess2.match_y(bar)
guess2.set_x(0, LEFT)
arrow = Arrow(bar.label, guess2, stroke_width=3, buff=SMALL_BUFF)
pattern_array2 = self.get_pattern_array(guess2, bucket, priors, n_shown=25)
prob_bars2 = self.get_prob_bars(pattern_array2.pattern_mobs, width_scalar=5)
h2_label = self.get_entropy_label(guess2, pattern_array2.distribution)
group = VGroup(
arrow, guess2, h2_label, pattern_array2, prob_bars2,
)
self.add(group, second_ents)
self.wait(1 / self.camera.frame_rate, ignore_presenter_mode=True)
if i in (0, 1) and j == 0:
self.wait(self.transition_time)
self.remove(group)
self.add(*group, second_ents)
self.wait()
# Consolidate
arrow, guess2, h2_label, pattern_array2, prob_bars2 = group
for line in pattern_array2.connecting_lines:
line.reverse_points()
h2_label.generate_target()
h2_label.target.scale(0.8)
h2_label.target.next_to(guess2, RIGHT)
guess2.set_color(YELLOW)
self.add(pattern_array2.connecting_lines, second_ents)
self.play(
MoveToTarget(h2_label),
Uncreate(pattern_array2.connecting_lines, lag_ratio=0.01),
LaggedStartMap(FadeOut, pattern_array2.pattern_mobs),
LaggedStartMap(FadeOut, pattern_array2.dot_parts),
LaggedStartMap(FadeOut, prob_bars2, scale=0.25),
run_time=self.transition_time
)
arrows.add(arrow)
second_guesses.add(guess2)
second_ents.add(h2_label)
# Show weighted sum
brace = Brace(VGroup(second_ents, pattern_array1), RIGHT)
label = brace.get_text("Compute a\nweighted average", buff=MED_SMALL_BUFF)
sum_parts = VGroup()
for bar, h_label in zip(prob_bars1, second_ents):
d0 = DecimalNumber(bar.prob, num_decimal_places=3)
d1 = h_label[1].copy()
d0.match_height(d1)
group = VGroup(d0, OldTex("\\cdot", font_size=24), d1)
group.generate_target()
group.target.arrange(RIGHT, buff=SMALL_BUFF)
group.target.next_to(brace, RIGHT)
group.target.match_y(bar)
sum_parts.add(group)
for part in group[:2]:
part.move_to(bar.label)
part.set_opacity(0)
self.play(
GrowFromCenter(brace),
Write(label)
)
self.wait()
self.play(
LaggedStartMap(MoveToTarget, sum_parts, run_time=2),
label.animate.scale(0.7).to_edge(DOWN),
)
self.wait()
def get_word_mob(self, word):
return Text(word, font="Consolas", font_size=36)
def get_pattern_array(self, word, possibilities, priors, n_shown=15):
weights = get_weights(possibilities, priors)
dist = get_pattern_distributions([word.text], possibilities, weights)[0]
indices = np.argsort(dist)[::-1]
patterns = np.arange(3**5)[indices]
patterns = patterns[:n_shown] # Only show non-zero possibilities
top_parts = VGroup(*(self.get_pattern_mob(p) for p in patterns[:n_shown]))
dot_parts = OldTex("\\vdots\\\\", "\\le 3^5 \\text{ patterns}\\\\", "\\vdots")
for prob, row in zip(dist[indices][:n_shown], top_parts):
row.prob = prob
stack = VGroup(*top_parts, *dot_parts)
dot_parts.match_width(stack[0])
stack.arrange(DOWN, buff=SMALL_BUFF)
stack.set_max_height(FRAME_HEIGHT - 1)
stack.next_to(word, RIGHT, buff=1.5)
# stack.set_y(0)
stack.shift_onto_screen(buff=MED_LARGE_BUFF)
pattern_mobs = top_parts
connecting_lines = VGroup(*(
self.get_connecting_line(word, row)
for row in pattern_mobs
))
result = VGroup(pattern_mobs, dot_parts, connecting_lines)
result.pattern_mobs = pattern_mobs
result.dot_parts = dot_parts
result.connecting_lines = connecting_lines
result.distribution = dist
return result
def get_pattern_mob(self, pattern, width=1.5):
result = Square().replicate(5)
result.arrange(RIGHT, buff=SMALL_BUFF)
result.set_stroke(WHITE, width=0.5)
for square, n in zip(result, pattern_to_int_list(pattern)):
square.set_fill(WordleScene.color_map[n], 1)
result.set_width(width)
result.pattern = pattern
return result
def get_connecting_line(self, mob1, mob2):
diff = mob2.get_left()[0] - mob1.get_right()[0]
return CubicBezier(
mob1.get_right() + SMALL_BUFF * RIGHT,
mob1.get_right() + RIGHT * diff / 2,
mob2.get_left() + LEFT * diff / 2,
mob2.get_left() + SMALL_BUFF * LEFT,
stroke_color=WHITE,
stroke_width=1,
)
def get_prob_bars(self, pattern_mobs, width_scalar=10):
result = VGroup()
for pattern_mob in pattern_mobs:
bar = Rectangle(
width=width_scalar * pattern_mob.prob,
height=pattern_mob.get_height(),
fill_color=self.prob_color,
fill_opacity=1,
stroke_width=0.5,
stroke_color=WHITE,
)
bar.next_to(pattern_mob, RIGHT, buff=SMALL_BUFF)
label = DecimalNumber(100 * pattern_mob.prob, num_decimal_places=1, unit="\\%")
# label = DecimalNumber(pattern_mob.prob, num_decimal_places=3)
label.set_height(bar.get_height() * 0.6)
label.next_to(bar, RIGHT, SMALL_BUFF)
bar.label = label
bar.add(label)
bar.prob = pattern_mob.prob
result.add(bar)
return result
def get_entropy_label(self, word_mob, distribution):
ent2 = entropy_of_distributions(distribution)
kw = dict(font_size=24)
h_label = VGroup(OldTex(f"E[I] = ", **kw), DecimalNumber(ent2, **kw))
h_label.set_color(self.entropy_color)
h_label.arrange(RIGHT, buff=SMALL_BUFF, aligned_edge=UP)
h_label.move_to(word_mob)
h_label.shift(0.5 * DOWN)
h_label.set_backstroke(width=8)
return h_label
def get_info_label(self, bar):
result = VGroup(
# DecimalNumber(bar.prob, num_decimal_places=3),
OldTex("\\log_2\\big( 1 / "),
DecimalNumber(bar.prob, num_decimal_places=3),
OldTex("\\big) = "),
# DecimalNumber(-bar.prob * math.log2(bar.prob), num_decimal_places=3)
DecimalNumber(-math.log2(bar.prob), num_decimal_places=3)
)
result.arrange(RIGHT, buff=SMALL_BUFF)
result.set_height(bar.get_height())
result.match_y(bar)
result.set_x(0, LEFT)
arrow = Arrow(bar.label.get_right(), result, stroke_width=2, buff=SMALL_BUFF)
result.add_to_back(arrow)
return result
class BestDoubleEntropies(Scene):
def construct(self):
pass
# Facts on theoretical possibilities:
# Best two-step entropy is slane: 5.7702 + 4.2435 = 10.014
# Given the start, with log2(2315) = 11.177 bits of entropy,
# this means an average uncertainty after two guesses of 1.163.
# This is akin to being down to 2.239 words
# In that case, there's a 1/2.239 = 0.4466 chance of getting it in 3
# Otherwise, 0.5534 chance of requiring at least 4
#
# Assuming best case scenarios, that out of the 2315 answers, you get:
# - 1 in 1
# - 273 in 2 with your encoded second guesses
# - Of the remaining 2041, you get 0.4466 * 2041 = 912 in 3
# - Of the remaining 1,129, all are in 4
# Average: (1 + 2 * 273 + 3 * 912 + 4 * 1129) / 2315 = 3.368
#
# But actually, number of 2's is (at most) 150, so we could update to:
# Average: (1 + 2 * 150 + 3 * 967 + 4 * 1197) / 2315 = 3.451
# More general formula
# p3 = 1 / 2**(np.log2(2315) - 10.014)
# (1 + 2 * n + 3 * p3 * (2315 - n - 1) + 4 * (1 - p3) * (2315 - n - 1)) / 2315
#
# Analyzing crane games, it looks like indeed, the average uncertainty
# at the third step is 1.2229, just slightly higher than the 1.163 above.
# In fact, for 'crane' the average shoudl be 11.177 - 9.9685 = 1.208
#
# game_data = json.load(open("/Users/grant/Dropbox/3Blue1Brown/data/wordle/crane_with_wordle_prior.json"))
# games = game_data["game_results"]
# reductions = [g['reductions'] for g in games]
# step3_state = [red[1] if len(red) > 1 else 1 for red in reductions]
# step3_bits = [math.log2(x) for x in step3_state]
# np.mean(step3_bits)
# Out: 1.2229
class TripleComparisonFrame(Scene):
def construct(self):
self.add(FullScreenRectangle())
squares = Square().replicate(3)
squares.stretch(0.9, 0)
squares.arrange(RIGHT, buff=0.1)
squares.set_width(FRAME_WIDTH - 0.5)
squares.set_stroke(WHITE, 2)
squares.set_fill(BLACK, 1)
squares.to_edge(DOWN, buff=1.5)
self.add(squares)
titles = VGroup(
OldTexText("V1: Just maximize\\\\entropy"),
OldTexText("V2: Incorporate word\\\\frequency data"),
OldTexText("V3: Use true wordle list\\\\(plus 1 or 2 other tricks)"),
)
titles.scale(0.75)
for title, square in zip(titles, squares):
title.next_to(square, UP)
self.add(titles)
self.embed()
class InformationLimit(WordleScene):
def construct(self):
# Setup
grid = self.grid
grid.set_height(4)
title = Text("Is there a fundamental limit?")
title.to_edge(UP, buff=MED_SMALL_BUFF)
line = Line(LEFT, RIGHT)
line.set_width(12)
line.set_stroke(WHITE, 1)
line.next_to(title, DOWN, buff=SMALL_BUFF)
self.add(title, line)
# Show wordle list
kw = dict(font_size=36)
left_title = Text("2,315 words, equally likely", **kw)
left_title.next_to(line, DOWN, buff=0.75)
left_title.to_edge(LEFT)
words = get_word_list(short=True)
word_mobs = self.get_grid_of_words(words, 18, 1, dots_index=-7)
word_mobs.set_height(5.5)
word_mobs.next_to(left_title, DOWN, aligned_edge=LEFT)
brace = Brace(word_mobs, RIGHT)
brace_label = VGroup(
OldTex("\\log_2(2{,}315)", "=", "11.17 \\text{ bits}", **kw),
Text("of uncertainty", **kw)
)
brace_label[0][1].rotate(PI / 2)
brace_label[0].arrange(DOWN, buff=MED_SMALL_BUFF)
brace_label.arrange(DOWN, aligned_edge=LEFT)
brace_label.set_color(TEAL)
brace_label[0][:2].set_color(WHITE)
brace_label.next_to(brace, RIGHT, SMALL_BUFF)
group = VGroup(word_mobs, left_title, brace_label)
grid.next_to(group, RIGHT)
self.play(
Write(left_title),
FadeIn(word_mobs, lag_ratio=0.1, run_time=3),
)
self.wait()
self.play(
GrowFromCenter(brace),
FadeIn(brace_label, 0.25 * RIGHT)
)
self.wait()
# Brute for search
all_words = get_word_list()
sample = random.sample(all_words, 180)
for word in sorted(sample):
self.add_word(word, wait_time_per_letter=0)
self.reveal_pattern(animate=False)
self.add_word(random.choice(all_words), wait_time_per_letter=0)
self.reveal_pattern(animate=False)
self.wait(1 / 30)
grid.words.set_submobjects([])
grid.pending_word.set_submobjects([])
grid.set_fill(BLACK, 0)
self.add_word("slane", wait_time_per_letter=0)
# Two step entropy
s_title = OldTexText(
"Maximum expected information\\\\",
"after first two guesses:",
**kw
)
s_title.match_y(left_title)
s_title.to_edge(RIGHT)
arrows = VGroup(*(
Arrow(
grid[i].get_right(),
grid[i + 1].get_right(),
buff=0,
path_arc=-(PI + 0.1),
width_to_tip_len=0.005
)
for i in (0, 1)
))
for arrow in arrows:
arrow.shift(0.1 * RIGHT)
arrows.space_out_submobjects(1.2)
EI_labels = VGroup(
OldTex("E[I_1] = 5.77", **kw),
OldTex("E[I_2] = 4.24", **kw),
)
EI_labels.set_color(BLUE)
for label, arrow in zip(EI_labels, arrows):
label.next_to(arrow, RIGHT, buff=SMALL_BUFF)
EI2_arrow = Vector(DR)
EI2_arrow.next_to(EI_labels[1].get_bottom(), DR, buff=SMALL_BUFF)
total_brace = Brace(EI_labels, RIGHT)
total_label = OldTexText("10.01 bits", **kw)
total_label.set_color(BLUE)
total_label.next_to(total_brace, RIGHT)
self.play(
Write(s_title),
ShowCreation(arrows),
)
self.play(LaggedStart(*(
FadeIn(EI_label, 0.25 * RIGHT)
for EI_label in EI_labels
)), lag_ratio=0.5)
self.play(
GrowFromCenter(total_brace),
FadeIn(total_label),
)
self.wait()
self.play(FadeIn(EI2_arrow))
self.play(FadeOut(EI2_arrow))
self.wait()
# Highlight third spot
row3 = grid[2].copy()
row3.set_stroke(RED, 3)
self.play(
grid.animate.fade(0.75),
FadeIn(row3),
)
self.wait()
# Best case words
best_case_words = Text(
"Best case scenario:\n"
"Down to ~1.16 bits of\n"
"uncertainty, on average",
**kw
)
best_case_words.get_part_by_text("~").match_y(
best_case_words.get_part_by_text("1.16")
)
best_case_words.next_to(grid[2], DR, MED_LARGE_BUFF)
self.play(Write(best_case_words))
self.wait()
class EndScreen(PatreonEndScreen):
CONFIG = {
"scroll_time": 30,
}
# Distribution animations
class ShowScoreDistribution(Scene):
data_file = "crane_with_wordle_prior.json"
axes_config = dict(
x_range=(0, 9),
y_range=(0, 1, 0.1),
width=8,
height=6,
)
weighted_sample = False
bar_count_font_size = 30
def construct(self):
axes = self.get_axes()
self.add(axes)
with open(os.path.join(DATA_DIR, "simulation_results", self.data_file)) as fp:
game_data = json.load(fp)
games = game_data["game_results"]
scores = [game["score"] for game in games]
bars = self.get_bars(axes, scores[:0])
mean_label = VGroup(
Text("Average score: "),
DecimalNumber(np.mean(scores), num_decimal_places=3)
)
mean_label.arrange(RIGHT, aligned_edge=UP)
mean_label.move_to(axes, UP)
self.add(mean_label)
grid = WordleScene.patterns_to_squares(6 * [0])
grid.set_fill(BLACK, 0)
grid.set_width(axes.get_width() / 4)
grid.move_to(axes, RIGHT)
grid.shift(0.5 * DOWN)
grid.words = VGroup()
grid.add(grid.words)
self.add(grid)
score_label = VGroup(
Text("Score: "),
Integer(0, edge_to_fix=LEFT),
)
score_label.scale(0.75)
score_label.arrange(RIGHT, aligned_edge=DOWN)
score_label.next_to(grid, UP, aligned_edge=LEFT)
self.add(score_label)
answer_label = VGroup(
Text("Answer: "),
Text(games[0]["answer"], font="Consolas")
)
answer_label.match_height(score_label)
answer_label.arrange(RIGHT)
answer_label.next_to(score_label, UP, aligned_edge=LEFT)
self.add(answer_label)
def a2n(alpha):
return integer_interpolate(0, len(scores), alpha)[0]
def update_bars(bars, alpha):
bars.set_submobjects(self.get_bars(axes, scores[:a2n(alpha) + 1]))
def update_mean_label(label, alpha):
label[1].set_value(np.mean(scores[:a2n(alpha) + 1]))
def update_grid(grid, alpha):
game = games[a2n(alpha)]
patterns = game["patterns"]
patterns.append(3**5 - 1)
grid.set_fill(BLACK, 0)
for pattern, row in zip(patterns, grid):
for square, key in zip(row, pattern_to_int_list(pattern)):
square.set_fill(WordleScene.color_map[key], 1)
try:
grid.words.set_submobjects([
Text(guess.upper(), font="Consolas")
for guess in (*game["guesses"], game["answer"])
])
except Exception:
return
for word, row in zip(grid.words, grid):
word.set_height(row.get_height() * 0.6)
for char, square in zip(word, row):
char.move_to(square)
def update_score_label(score_label, alpha):
score = games[a2n(alpha)]["score"]
score_label[1].set_value(score)
def update_answer_label(answer_label, alpha):
answer = games[a2n(alpha)]["answer"]
new_text = Text(answer, font="Consolas")
new_text.scale(0.75)
new_text.move_to(answer_label[1], LEFT)
low_y = new_text[np.argmin([c.get_height() for c in new_text])].get_bottom()[1]
new_text.shift((answer_label[0].get_bottom()[1] - low_y) * UP)
answer_label.replace_submobject(1, new_text)
return answer_label
self.play(
UpdateFromAlphaFunc(bars, update_bars),
UpdateFromAlphaFunc(mean_label, update_mean_label),
UpdateFromAlphaFunc(grid, update_grid),
UpdateFromAlphaFunc(score_label, update_score_label),
UpdateFromAlphaFunc(answer_label, update_answer_label),
run_time=20,
rate_func=linear,
)
update_bars(bars, 1)
self.wait()
self.remove(grid, score_label, answer_label)
def get_axes(self):
axes = Axes(**self.axes_config)
x_axis, y_axis = axes.x_axis, axes.y_axis
y_axis.add_numbers(num_decimal_places=1)
x_axis.add_numbers()
x_axis.numbers.shift(x_axis.unit_size * LEFT / 2)
x_label = Text("Score", font_size=24)
x_label.next_to(x_axis.get_end(), RIGHT)
x_axis.add(x_label)
return axes
def get_bars(self, axes, scores):
scores = np.array(scores)
buckets = np.array([
(scores == n + 1).sum()
for n in np.arange(*axes.x_range)
])
props = buckets / buckets.sum()
bars = VGroup(*(
self.get_bar(axes, n + 1, prop)
for n, prop in enumerate(props)
))
colors = color_gradient([BLUE, YELLOW, RED], 8)
for bar, color in zip(bars, colors):
bar.set_fill(color, 1)
bars.set_stroke(WHITE, 1)
for bar, count in zip(bars, buckets):
bar.add(self.get_bar_count(bar, count))
return VGroup(bars)
def get_bar(self, axes, score, proportion):
bar = Rectangle(
width=axes.x_axis.unit_size,
height=axes.y_axis.unit_size * proportion,
)
bar.set_fill(BLUE, 1)
bar.set_stroke(WHITE, 1)
bar.move_to(axes.c2p(score, 0), DR)
return bar
def get_bar_count(self, bar, count):
result = Integer(count, font_size=self.bar_count_font_size)
result.set_max_width(bar.get_width() * 0.8)
result.next_to(bar, UP, SMALL_BUFF)
if count == 0:
result.set_opacity(0)
return result
class SimulatedGamesUniformPriorDist(ShowScoreDistribution):
data_file = "tares_with_uniform_prior.json"
class SimulatedGamesFreqBasedPriorDist(ShowScoreDistribution):
data_file = "tares_with_freq_prior.json"
class SimulatedGamesWordleBasedPriorDist(ShowScoreDistribution):
data_file = "soare_with_wordle_prior.json"
class SimulatedGamesWordleBasedPriorCraneStartDist(ShowScoreDistribution):
data_file = "crane_with_wordle_prior.json"
class SimulatedGamesCraneHardModeDist(ShowScoreDistribution):
data_file = "crane_hard_mode.json"
class SimulatedGamesWordleBasedPriorExcludeSeenWordsDist(ShowScoreDistribution):
data_file = "crane_with_wordle_prior_exclude_seen.json"
class SimulatedGamesFreqBasedPriorExcludeSeenWordsDist(ShowScoreDistribution):
data_file = "tares_with_freq_prior_exclude_seen.json"
class ThinV1Stats(SimulatedGamesUniformPriorDist):
axes_config = dict(
x_range=(0, 8),
y_range=(0, 1, 0.1),
width=5,
height=6,
)
bar_count_font_size = 24
class ThinV2Stats(SimulatedGamesFreqBasedPriorDist):
axes_config = ThinV1Stats.axes_config
bar_count_font_size = 24
class ThinV3Stats(SimulatedGamesWordleBasedPriorCraneStartDist):
axes_config = ThinV1Stats.axes_config
bar_count_font_size = 24
class GenericWrapper(VideoWrapper):
pass
# Thumbnail
class Thumbnail(Scene):
def construct(self):
# Grid
answer = "aging"
guesses = ["crane", "tousy", answer]
patterns = [get_pattern(guess, answer) for guess in guesses]
rows = WordleScene.patterns_to_squares(
patterns, color_map=[GREY_D, YELLOW, GREEN_E]
)
rows.set_stroke(width=0)
rows.set_width(0.5 * FRAME_WIDTH)
rows.to_edge(DOWN, buff=1.0)
rows.set_gloss(0.4)
self.add(rows)
# Words in grid (probbaly don't include)
words = VGroup()
for guess, row in zip(guesses, rows):
word = Text(guess.upper(), font="Consolas")
word.set_height(0.6 * row.get_height())
for char, square in zip(word, row):
char.move_to(square)
words.add(word)
# self.add(words)
# Title
title = Text(
"Best opener: CRANE",
font_size=100,
font="Consolas",
t2c={"crane": GREEN}
)
title.to_edge(UP, buff=0.75)
self.add(title)
self.rows = rows
self.title = title
|
|
from manim_imports_ext import *
from tqdm import tqdm as ProgressDisplay
from scipy.stats import entropy
MISS = np.uint8(0)
MISPLACED = np.uint8(1)
EXACT = np.uint8(2)
DATA_DIR = os.path.join(
os.path.dirname(os.path.realpath(__file__)),
"data",
)
SHORT_WORD_LIST_FILE = os.path.join(DATA_DIR, "possible_words.txt")
LONG_WORD_LIST_FILE = os.path.join(DATA_DIR, "allowed_words.txt")
WORD_FREQ_FILE = os.path.join(DATA_DIR, "wordle_words_freqs_full.txt")
WORD_FREQ_MAP_FILE = os.path.join(DATA_DIR, "freq_map.json")
SECOND_GUESS_MAP_FILE = os.path.join(DATA_DIR, "second_guess_map.json")
PATTERN_MATRIX_FILE = os.path.join(DATA_DIR, "pattern_matrix.npy")
ENT_SCORE_PAIRS_FILE = os.path.join(DATA_DIR, "ent_score_pairs.json")
# To store the large grid of patterns at run time
PATTERN_GRID_DATA = dict()
def safe_log2(x):
return math.log2(x) if x > 0 else 0
# Reading from files
def get_word_list(short=False):
result = []
file = SHORT_WORD_LIST_FILE if short else LONG_WORD_LIST_FILE
with open(file) as fp:
result.extend([word.strip() for word in fp.readlines()])
return result
def get_word_frequencies(regenerate=False):
if os.path.exists(WORD_FREQ_MAP_FILE) or regenerate:
with open(WORD_FREQ_MAP_FILE) as fp:
result = json.load(fp)
return result
# Otherwise, regenerate
freq_map = dict()
with open(WORD_FREQ_FILE) as fp:
for line in fp.readlines():
pieces = line.split(' ')
word = pieces[0]
freqs = [
float(piece.strip())
for piece in pieces[1:]
]
freq_map[word] = np.mean(freqs[-5:])
with open(WORD_FREQ_MAP_FILE, 'w') as fp:
json.dump(freq_map, fp)
return freq_map
def get_frequency_based_priors(n_common=3000, width_under_sigmoid=10):
"""
We know that that list of wordle answers was curated by some human
based on whether they're sufficiently common. This function aims
to associate each word with the likelihood that it would actually
be selected for the final answer.
Sort the words by frequency, then apply a sigmoid along it.
"""
freq_map = get_word_frequencies()
words = np.array(list(freq_map.keys()))
freqs = np.array([freq_map[w] for w in words])
arg_sort = freqs.argsort()
sorted_words = words[arg_sort]
# We want to imagine taking this sorted list, and putting it on a number
# line so that it's length is 10, situating it so that the n_common most common
# words are positive, then applying a sigmoid
x_width = width_under_sigmoid
c = x_width * (-0.5 + n_common / len(words))
xs = np.linspace(c - x_width / 2, c + x_width / 2, len(words))
priors = dict()
for word, x in zip(sorted_words, xs):
priors[word] = sigmoid(x)
return priors
def get_true_wordle_prior():
words = get_word_list()
short_words = get_word_list(short=True)
return dict(
(w, int(w in short_words))
for w in words
)
# Generating color patterns between strings, etc.
def words_to_int_arrays(words):
return np.array([[ord(c)for c in w] for w in words], dtype=np.uint8)
def generate_pattern_matrix(words1, words2):
"""
A pattern for two words represents the wordle-similarity
pattern (grey -> 0, yellow -> 1, green -> 2) but as an integer
between 0 and 3^5. Reading this integer in ternary gives the
associated pattern.
This function computes the pairwise patterns between two lists
of words, returning the result as a grid of hash values. Since
this can be time-consuming, many operations that can be are vectorized
(perhaps at the expense of easier readibility), and the the result
is saved to file so that this only needs to be evaluated once, and
all remaining pattern matching is a lookup.
"""
# Number of letters/words
nl = len(words1[0])
nw1 = len(words1) # Number of words
nw2 = len(words2) # Number of words
# Convert word lists to integer arrays
word_arr1, word_arr2 = map(words_to_int_arrays, (words1, words2))
# equality_grid keeps track of all equalities between all pairs
# of letters in words. Specifically, equality_grid[a, b, i, j]
# is true when words[i][a] == words[b][j]
equality_grid = np.zeros((nw1, nw2, nl, nl), dtype=bool)
for i, j in it.product(range(nl), range(nl)):
equality_grid[:, :, i, j] = np.equal.outer(word_arr1[:, i], word_arr2[:, j])
# full_pattern_matrix[a, b] should represent the 5-color pattern
# for guess a and answer b, with 0 -> grey, 1 -> yellow, 2 -> green
full_pattern_matrix = np.zeros((nw1, nw2, nl), dtype=np.uint8)
# Green pass
for i in range(nl):
matches = equality_grid[:, :, i, i].flatten() # matches[a, b] is true when words[a][i] = words[b][i]
full_pattern_matrix[:, :, i].flat[matches] = EXACT
for k in range(nl):
# If it's a match, mark all elements associated with
# that letter, both from the guess and answer, as covered.
# That way, it won't trigger the yellow pass.
equality_grid[:, :, k, i].flat[matches] = False
equality_grid[:, :, i, k].flat[matches] = False
# Yellow pass
for i, j in it.product(range(nl), range(nl)):
matches = equality_grid[:, :, i, j].flatten()
full_pattern_matrix[:, :, i].flat[matches] = MISPLACED
for k in range(nl):
# Similar to above, we want to mark this letter
# as taken care of, both for answer and guess
equality_grid[:, :, k, j].flat[matches] = False
equality_grid[:, :, i, k].flat[matches] = False
# Rather than representing a color pattern as a lists of integers,
# store it as a single integer, whose ternary representations corresponds
# to that list of integers.
pattern_matrix = np.dot(
full_pattern_matrix,
(3**np.arange(nl)).astype(np.uint8)
)
return pattern_matrix
def generate_full_pattern_matrix():
words = get_word_list()
pattern_matrix = generate_pattern_matrix(words, words)
# Save to file
np.save(PATTERN_MATRIX_FILE, pattern_matrix)
return pattern_matrix
def get_pattern_matrix(words1, words2):
if not PATTERN_GRID_DATA:
if not os.path.exists(PATTERN_MATRIX_FILE):
log.info("\n".join([
"Generating pattern matrix. This takes a minute, but",
"the result will be saved to file so that it only",
"needs to be computed once.",
]))
generate_full_pattern_matrix()
PATTERN_GRID_DATA['grid'] = np.load(PATTERN_MATRIX_FILE)
PATTERN_GRID_DATA['words_to_index'] = dict(zip(
get_word_list(), it.count()
))
full_grid = PATTERN_GRID_DATA['grid']
words_to_index = PATTERN_GRID_DATA['words_to_index']
indices1 = [words_to_index[w] for w in words1]
indices2 = [words_to_index[w] for w in words2]
return full_grid[np.ix_(indices1, indices2)]
def get_pattern(guess, answer):
if PATTERN_GRID_DATA:
saved_words = PATTERN_GRID_DATA['words_to_index']
if guess in saved_words and answer in saved_words:
return get_pattern_matrix([guess], [answer])[0, 0]
return generate_pattern_matrix([guess], [answer])[0, 0]
def pattern_from_string(pattern_string):
return sum((3**i) * int(c) for i, c in enumerate(pattern_string))
def pattern_to_int_list(pattern):
result = []
curr = pattern
for x in range(5):
result.append(curr % 3)
curr = curr // 3
return result
def pattern_to_string(pattern):
d = {MISS: "⬛", MISPLACED: "🟨", EXACT: "🟩"}
return "".join(d[x] for x in pattern_to_int_list(pattern))
def patterns_to_string(patterns):
return "\n".join(map(pattern_to_string, patterns))
def get_possible_words(guess, pattern, word_list):
all_patterns = get_pattern_matrix([guess], word_list).flatten()
return list(np.array(word_list)[all_patterns == pattern])
def get_word_buckets(guess, possible_words):
buckets = [[] for x in range(3**5)]
hashes = get_pattern_matrix([guess], possible_words).flatten()
for index, word in zip(hashes, possible_words):
buckets[index].append(word)
return buckets
# Functions associated with entropy calculation
def get_weights(words, priors):
frequencies = np.array([priors[word] for word in words])
total = frequencies.sum()
if total == 0:
return np.zeros(frequencies.shape)
return frequencies / total
def get_pattern_distributions(allowed_words, possible_words, weights):
"""
For each possible guess in allowed_words, this finds the probability
distribution across all of the 3^5 wordle patterns you could see, assuming
the possible answers are in possible_words with associated probabilities
in weights.
It considers the pattern hash grid between the two lists of words, and uses
that to bucket together words from possible_words which would produce
the same pattern, adding together their corresponding probabilities.
"""
pattern_matrix = get_pattern_matrix(allowed_words, possible_words)
n = len(allowed_words)
distributions = np.zeros((n, 3**5))
n_range = np.arange(n)
for j, prob in enumerate(weights):
distributions[n_range, pattern_matrix[:, j]] += prob
return distributions
def entropy_of_distributions(distributions, atol=1e-12):
axis = len(distributions.shape) - 1
return entropy(distributions, base=2, axis=axis)
def get_entropies(allowed_words, possible_words, weights):
if weights.sum() == 0:
return np.zeros(len(allowed_words))
distributions = get_pattern_distributions(allowed_words, possible_words, weights)
return entropy_of_distributions(distributions)
def max_bucket_size(guess, possible_words, weights):
dist = get_pattern_distributions([guess], possible_words, weights)
return dist.max()
def words_to_max_buckets(possible_words, weights):
return dict(
(word, max_bucket_size(word, possible_words, weights))
for word in ProgressDisplay(possible_words)
)
words_and_maxes = list(w2m.items())
words_and_maxes.sort(key=lambda t: t[1])
words_and_maxes[:-20:-1]
def get_bucket_sizes(allowed_words, possible_words):
"""
Returns a (len(allowed_words), 243) shape array reprenting the size of
word buckets associated with each guess in allowed_words
"""
weights = np.ones(len(possible_words))
return get_pattern_distributions(allowed_words, possible_words, weights)
def get_bucket_counts(allowed_words, possible_words):
"""
Returns the number of separate buckets that each guess in allowed_words
would separate possible_words into
"""
bucket_sizes = get_bucket_sizes(allowed_words, possible_words)
return (bucket_sizes > 0).sum(1)
# Functions to analyze second guesses
def get_average_second_step_entropies(first_guesses, allowed_second_guesses, possible_words, priors):
result = []
weights = get_weights(possible_words, priors)
if weights.sum() == 0:
return np.zeros(len(first_guesses))
distributions = get_pattern_distributions(first_guesses, possible_words, weights)
for first_guess, dist in ProgressDisplay(list(zip(first_guesses, distributions)), leave=False, desc="Searching 2nd step entropies"):
word_buckets = get_word_buckets(first_guess, possible_words)
# List of maximum entropies you could achieve in
# the second step for each pattern you might see
# after this setp
ents2 = np.array([
get_entropies(
allowed_words=allowed_second_guesses,
possible_words=bucket,
weights=get_weights(bucket, priors)
).max()
for bucket in word_buckets
])
# Multiply each such maximal entropy by the corresponding
# probability of falling into that bucket
result.append(np.dot(ents2, dist))
return np.array(result)
# Solvers
def get_guess_values_array(allowed_words, possible_words, priors, look_two_ahead=False):
weights = get_weights(possible_words, priors)
ents1 = get_entropies(allowed_words, possible_words, weights)
probs = np.array([
0 if word not in possible_words else weights[possible_words.index(word)]
for word in allowed_words
])
if look_two_ahead:
# Look two steps out, but restricted to where second guess is
# amoung the remaining possible words
ents2 = np.zeros(ents1.shape)
top_indices = np.argsort(ents1)[-250:]
ents2[top_indices] = get_average_second_step_entropies(
first_guesses=np.array(allowed_words)[top_indices],
allowed_second_guesses=allowed_words,
possible_words=possible_words,
priors=priors
)
return np.array([ents1, ents2, probs])
else:
return np.array([ents1, probs])
def entropy_to_expected_score(ent):
"""
Based on a regression associating entropies with typical scores
from that point forward in simulated games, this function returns
what the expected number of guesses required will be in a game where
there's a given amount of entropy in the remaining possibilities.
"""
# Assuming you can definitely get it in the next guess,
# this is the expected score
min_score = 2**(-ent) + 2 * (1 - 2**(-ent))
# To account for the likely uncertainty after the next guess,
# and knowing that entropy of 11.5 bits seems to have average
# score of 3.5, we add a line to account
# we add a line which connects (0, 0) to (3.5, 11.5)
return min_score + 1.5 * ent / 11.5
def get_expected_scores(allowed_words, possible_words, priors,
look_two_ahead=False,
n_top_candidates_for_two_step=25,
):
# Currenty entropy of distribution
weights = get_weights(possible_words, priors)
H0 = entropy_of_distributions(weights)
H1s = get_entropies(allowed_words, possible_words, weights)
word_to_weight = dict(zip(possible_words, weights))
probs = np.array([word_to_weight.get(w, 0) for w in allowed_words])
# If this guess is the true answer, score is 1. Otherwise, it's 1 plus
# the expected number of guesses it will take after getting the corresponding
# amount of information.
expected_scores = probs + (1 - probs) * (1 + entropy_to_expected_score(H0 - H1s))
if not look_two_ahead:
return expected_scores
# For the top candidates, refine the score by looking two steps out
# This is currently quite slow, and could be optimized to be faster.
# But why?
sorted_indices = np.argsort(expected_scores)
allowed_second_guesses = get_word_list()
expected_scores += 1 # Push up the rest
for i in ProgressDisplay(sorted_indices[:n_top_candidates_for_two_step], leave=False):
guess = allowed_words[i]
H1 = H1s[i]
dist = get_pattern_distributions([guess], possible_words, weights)[0]
buckets = get_word_buckets(guess, possible_words)
second_guesses = [
optimal_guess(allowed_second_guesses, bucket, priors, look_two_ahead=False)
for bucket in buckets
]
H2s = [
get_entropies([guess2], bucket, get_weights(bucket, priors))[0]
for guess2, bucket in zip(second_guesses, buckets)
]
prob = word_to_weight.get(guess, 0)
expected_scores[i] = sum((
# 1 times Probability guess1 is correct
1 * prob,
# 2 times probability guess2 is correct
2 * (1 - prob) * sum(
p * word_to_weight.get(g2, 0)
for p, g2 in zip(dist, second_guesses)
),
# 2 plus expected score two steps from now
(1 - prob) * (2 + sum(
p * (1 - word_to_weight.get(g2, 0)) * entropy_to_expected_score(H0 - H1 - H2)
for p, g2, H2 in zip(dist, second_guesses, H2s)
))
))
return expected_scores
def get_score_lower_bounds(allowed_words, possible_words):
"""
Assuming a uniform distribution on how likely each element
of possible_words is, this gives the a lower boudn on the
possible score for each word in allowed_words
"""
bucket_counts = get_bucket_counts(allowed_words, possible_words)
N = len(possible_words)
# Probabilities of getting it in 1
p1s = np.array([w in possible_words for w in allowed_words]) / N
# Probabilities of getting it in 2
p2s = bucket_counts / N - p1s
# Otherwise, assume it's gotten in 3 (which is optimistics)
p3s = 1 - bucket_counts / N
return p1s + 2 * p2s + 3 * p3s
def optimal_guess(allowed_words, possible_words, priors,
look_two_ahead=False,
optimize_for_uniform_distribution=False,
purely_maximize_information=False,
):
if purely_maximize_information:
if len(possible_words) == 1:
return possible_words[0]
weights = get_weights(possible_words, priors)
ents = get_entropies(allowed_words, possible_words, weights)
return allowed_words[np.argmax(ents)]
# Just experimenting here...
if optimize_for_uniform_distribution:
expected_scores = get_score_lower_bounds(
allowed_words, possible_words
)
else:
expected_scores = get_expected_scores(
allowed_words, possible_words, priors,
look_two_ahead=look_two_ahead
)
return allowed_words[np.argmin(expected_scores)]
def brute_force_optimal_guess(all_words, possible_words, priors, n_top_picks=10, display_progress=False):
if len(possible_words) == 0:
# Doesn't matter what to return in this case, so just default to first word in list.
return all_words[0]
# For the suggestions with the top expected scores, just
# actually play the game out from this point to see what
# their actual scores are, and minimize.
expected_scores = get_score_lower_bounds(all_words, possible_words)
top_choices = [all_words[i] for i in np.argsort(expected_scores)[:n_top_picks]]
true_average_scores = []
if display_progress:
iterable = ProgressDisplay(
top_choices,
desc=f"Possibilities: {len(possible_words)}",
leave=False
)
else:
iterable = top_choices
for next_guess in iterable:
scores = []
for answer in possible_words:
score = 1
possibilities = list(possible_words)
guess = next_guess
while guess != answer:
possibilities = get_possible_words(
guess, get_pattern(guess, answer),
possibilities,
)
# Make recursive? If so, we'd want to keep track of
# the next_guess map and pass it down in the recursive
# subcalls
guess = optimal_guess(
all_words, possibilities, priors,
optimize_for_uniform_distribution=True
)
score += 1
scores.append(score)
true_average_scores.append(np.mean(scores))
return top_choices[np.argmin(true_average_scores)]
# Run simulated wordle games
def get_two_step_score_lower_bound(first_guess, allowed_words, possible_words):
"""
Useful to prove what the minimum possible average score could be
for a given initial guess
"""
N = len(possible_words)
buckets = get_word_buckets(first_guess, possible_words)
min_score = 0
for bucket in buckets:
if len(bucket) == 0:
continue
lower_bounds = get_score_lower_bounds(allowed_words, bucket)
min_score += (len(bucket) / N) * lower_bounds.min()
p = (1 / len(possible_words)) * (first_guess in possible_words)
return p + (1 - p) * (1 + min_score)
def find_top_scorers(n_top_candidates=100, quiet=True, file_ext="", **kwargs):
# Run find_best_two_step_entropy first
file = os.path.join(get_directories()["data"], "wordle", "best_double_entropies.json")
with open(file) as fp:
double_ents = json.load(fp)
answers = get_word_list(short=True)
priors = get_true_wordle_prior()
guess_to_score = {}
guess_to_dist = {}
for row in ProgressDisplay(double_ents[:n_top_candidates]):
first_guess = row[0]
result, decision_map = simulate_games(
first_guess, priors=priors,
optimize_for_uniform_distribution=True,
quiet=quiet,
**kwargs,
)
average = result["average_score"]
total = int(np.round(average * len(answers)))
guess_to_score[first_guess] = total
guess_to_dist[first_guess] = result["score_distribution"]
top_scorers = sorted(list(guess_to_score.keys()), key=lambda w: guess_to_score[w])
result = [[w, guess_to_score[w], guess_to_dist[w]] for w in top_scorers]
file = os.path.join(
get_directories()["data"], "wordle",
"best_scores" + file_ext + ".json",
)
with open(file, 'w') as fp:
json.dump(result, fp)
return result
def find_best_two_step_entropy():
words = get_word_list()
answers = get_word_list(short=True)
priors = get_true_wordle_prior()
ents = get_entropies(words, answers, get_weights(answers, priors))
sorted_indices = np.argsort(ents)
top_candidates = np.array(words)[sorted_indices[:-250:-1]]
top_ents = ents[sorted_indices[:-250:-1]]
ent_file = os.path.join(get_directories()["data"], "wordle", "best_entropies.json")
with open(ent_file, 'w') as fp:
json.dump([[tc, te] for tc, te in zip(top_candidates, top_ents)], fp)
ents2 = get_average_second_step_entropies(
top_candidates, words, answers, priors,
)
total_ents = top_ents + ents2
sorted_indices2 = np.argsort(total_ents)
double_ents = [
[top_candidates[i], top_ents[i], ents2[i]]
for i in sorted_indices2[::-1]
]
ent2_file = os.path.join(get_directories()["data"], "wordle", "best_double_entropies.json")
with open(ent2_file, 'w') as fp:
json.dump(double_ents, fp)
return double_ents
def find_smallest_second_guess_buckets(n_top_picks=100):
all_words = get_word_list()
possibilities = get_word_list(short=True)
priors = get_true_wordle_prior()
weights = get_weights(possibilities, priors)
dists = get_pattern_distributions(all_words, possibilities, weights)
sorted_indices = np.argsort((dists**2).sum(1))
top_indices = sorted_indices[:n_top_picks]
top_picks = np.array(all_words)[top_indices]
top_dists = dists[top_indices]
# Figure out the average number of matching words there will
# be after two steps of game play
avg_ts_buckets = []
for first_guess, dist in ProgressDisplay(list(zip(top_picks, top_dists))):
buckets = get_word_buckets(first_guess, possibilities)
avg_ts_bucket = 0
for p, bucket in zip(dist, buckets):
weights = get_weights(bucket, priors)
sub_dists = get_pattern_distributions(all_words, bucket, weights)
min_ts_bucket = len(bucket) * (sub_dists**2).sum(1).min()
avg_ts_bucket += p * min_ts_bucket
avg_ts_buckets.append(avg_ts_bucket)
result = []
for j in np.argsort(avg_ts_buckets):
i = top_indices[j]
result.append((
# Word
all_words[i],
# Average bucket size after first guess
len(possibilities) * (dists[i]**2).sum(),
# Average bucket size after second, with optimal
# play.
avg_ts_buckets[j],
))
return result
def get_optimal_second_guess_map(first_guess, n_top_picks=10, regenerate=False):
with open(SECOND_GUESS_MAP_FILE) as fp:
all_sgms = json.load(fp)
if first_guess in all_sgms and not regenerate:
return all_sgms[first_guess]
log.info("\n".join([
f"Generating optimal second guess map for {first_guess}.",
"This involves brute forcing many simulations",
"so can take a little while."
]))
sgm = [""] * 3**5
all_words = get_word_list()
wordle_answers = get_word_list(short=True)
priors = get_true_wordle_prior()
buckets = get_word_buckets(first_guess, wordle_answers)
for pattern, bucket in ProgressDisplay(list(enumerate(buckets)), leave=False):
sgm[pattern] = brute_force_optimal_guess(
all_words, bucket, priors,
n_top_picks=n_top_picks,
display_progress=True
)
# Save to file
with open(SECOND_GUESS_MAP_FILE) as fp:
all_sgms = json.load(fp)
all_sgms[first_guess] = sgm
with open(SECOND_GUESS_MAP_FILE, 'w') as fp:
json.dump(all_sgms, fp)
return sgm
def gather_entropy_to_score_data(first_guess="crane", priors=None):
words = get_word_list()
answers = get_word_list(short=True)
if priors is None:
priors = get_true_wordle_prior()
# List of entropy/score pairs
ent_score_pairs = []
for answer in ProgressDisplay(answers):
score = 1
possibilities = list(filter(lambda w: priors[w] > 0, words))
guess = first_guess
guesses = []
entropies = []
while True:
guesses.append(guess)
weights = get_weights(possibilities, priors)
entropies.append(entropy_of_distributions(weights))
if guess == answer:
break
possibilities = get_possible_words(
guess, get_pattern(guess, answer), possibilities
)
guess = optimal_guess(words, possibilities, priors)
score += 1
for sc, ent in zip(it.count(1), reversed(entropies)):
ent_score_pairs.append((ent, sc))
with open(ENT_SCORE_PAIRS_FILE, 'w') as fp:
json.dump(ent_score_pairs, fp)
return ent_score_pairs
def simulate_games(first_guess=None,
priors=None,
look_two_ahead=False,
optimize_for_uniform_distribution=False,
second_guess_map=None,
exclude_seen_words=False,
test_set=None,
shuffle=False,
hard_mode=False,
purely_maximize_information=False,
brute_force_optimize=False,
brute_force_depth=10,
results_file=None,
next_guess_map_file=None,
quiet=False,
):
all_words = get_word_list(short=False)
short_word_list = get_word_list(short=True)
if first_guess is None:
first_guess = optimal_guess(
all_words, all_words, priors,
**choice_config
)
if priors is None:
priors = get_frequency_based_priors()
if test_set is None:
test_set = short_word_list
if shuffle:
random.shuffle(test_set)
seen = set()
# Function for choosing the next guess, with a dict to cache
# and reuse results that are seen multiple times in the sim
next_guess_map = {}
def get_next_guess(guesses, patterns, possibilities):
phash = "".join(
str(g) + "".join(map(str, pattern_to_int_list(p)))
for g, p in zip(guesses, patterns)
)
if second_guess_map is not None and len(patterns) == 1:
next_guess_map[phash] = second_guess_map[patterns[0]]
if phash not in next_guess_map:
choices = all_words
if hard_mode:
for guess, pattern in zip(guesses, patterns):
choices = get_possible_words(guess, pattern, choices)
if brute_force_optimize:
next_guess_map[phash] = brute_force_optimal_guess(
choices, possibilities, priors,
n_top_picks=brute_force_depth,
)
else:
next_guess_map[phash] = optimal_guess(
choices, possibilities, priors,
look_two_ahead=look_two_ahead,
purely_maximize_information=purely_maximize_information,
optimize_for_uniform_distribution=optimize_for_uniform_distribution,
)
return next_guess_map[phash]
# Go through each answer in the test set, play the game,
# and keep track of the stats.
scores = np.zeros(0, dtype=int)
game_results = []
for answer in ProgressDisplay(test_set, leave=False, desc=" Trying all wordle answers"):
guesses = []
patterns = []
possibility_counts = []
possibilities = list(filter(lambda w: priors[w] > 0, all_words))
if exclude_seen_words:
possibilities = list(filter(lambda w: w not in seen, possibilities))
score = 1
guess = first_guess
while guess != answer:
pattern = get_pattern(guess, answer)
guesses.append(guess)
patterns.append(pattern)
possibilities = get_possible_words(guess, pattern, possibilities)
possibility_counts.append(len(possibilities))
score += 1
guess = get_next_guess(guesses, patterns, possibilities)
# Accumulate stats
scores = np.append(scores, [score])
score_dist = [
int((scores == i).sum())
for i in range(1, scores.max() + 1)
]
total_guesses = scores.sum()
average = scores.mean()
seen.add(answer)
game_results.append(dict(
score=int(score),
answer=answer,
guesses=guesses,
patterns=list(map(int, patterns)),
reductions=possibility_counts,
))
# Print outcome
if not quiet:
message = "\n".join([
"",
f"Score: {score}",
f"Answer: {answer}",
f"Guesses: {guesses}",
f"Reductions: {possibility_counts}",
*patterns_to_string((*patterns, 3**5 - 1)).split("\n"),
*" " * (6 - len(patterns)),
f"Distribution: {score_dist}",
f"Total guesses: {total_guesses}",
f"Average: {average}",
*" " * 2,
])
if answer is not test_set[0]:
# Move cursor back up to the top of the message
n = len(message.split("\n")) + 1
print(("\033[F\033[K") * n)
else:
print("\r\033[K\n")
print(message)
final_result = dict(
score_distribution=score_dist,
total_guesses=int(total_guesses),
average_score=float(scores.mean()),
game_results=game_results,
)
# Save results
for obj, file in [(final_result, results_file), (next_guess_map, next_guess_map_file)]:
if file:
path = os.path.join(DATA_DIR, "simulation_results", file)
with open(path, 'w') as fp:
json.dump(obj, fp)
return final_result, next_guess_map
if __name__ == "__main__":
first_guess = "salet"
results, decision_map = simulate_games(
first_guess=first_guess,
priors=get_true_wordle_prior(),
optimize_for_uniform_distribution=True,
# shuffle=True,
# brute_force_optimize=True,
# hard_mode=True,
)
|
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"from manim_imports_ext import *\nfrom _2022.wordle.scenes import *\n\n\nclass HeresTheThing(Teacher(...TRUNCATED) |
|
"from manim_imports_ext import *\nimport sympy\n\n\nPRIME_COLOR = YELLOW\n\n\ndef get_primes(max_n=1(...TRUNCATED) |
|
"from manim_imports_ext import *\n\n\nEQUATOR_STYLE = dict(stroke_color=TEAL, stroke_width=2)\n\n\nd(...TRUNCATED) |
|
"from manim_imports_ext import *\nfrom _2022.piano.fourier_animations import DecomposeAudioSegment\n(...TRUNCATED) |
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