space and games

February 8, 2012

Large-Middle-Small: A Kanji Word Block

Filed under: General — Peter de Blanc @ 9:02 am

I’m investigating the Japanese language.

A Kanji word block is a 3 x 3 block of Kanji characters in which each row and each column forms a word, and no character appears more than once. Thus the word block contains six words in all. Using Jim Breen’s WWWJDIC as my reference, I found that there are 12 possible word blocks. Here is one:


ダイ / おお

Large

チュウ

Middle

ショウ / こ

Small
だいちゅうしょう
L/M/S; Clothing Sizes

ニン

Person

ケン / カン

Space

がた

Model
にんげんがた
Humanoid

スウ

Number



Value



Change
すうちか
Numeric Conversion

おおにんずう
Large Number
of People

ちゅうかんち
Median

こがたか
Miniaturization

So a person-space-model is a humanoid, a number-value-change is a numeric conversion, and so on.

February 3, 2012

The Item and the Thing

Filed under: General — Peter de Blanc @ 2:56 am

We show Sam the item.
“What is this?” we ask.
“I don’t know,” says Sam.
We say, “It is the item.”
Again, we ask Sam, “what is this?”
“The item,” Sam repeats.
Sam has received his first lesson.

We show Sam the item.
“What is this?” we ask.
“I’m not sure,” says Sam, “but I think it is the item.”
“Of course it is the item! Why are you not sure?”
Sam has no reply.

The thing is very similar to the item.
We show Sam the thing.
“What is this?” we ask.
“It is the item!” Sam shouts confidently.
“It is not the item! It is the thing!”
Was Sam’s earlier hesitation justified?

We show Lisa the item.
“What is this?” we ask.
“I don’t know,” says Lisa.
We say, “It is the item.”
Again, we ask Lisa, “what is this?”
“The item,” Lisa repeats.
We show Lisa the thing.
“What is this?” we ask.
“I don’t know,” says Lisa.
We say, “It is the thing.”
Again, we ask Lisa, “what is this?”
“The thing,” Lisa repeats.
Lisa has received two lessons.

December 20, 2011

Google Books 1-grams

Filed under: General — Peter de Blanc @ 11:53 pm

Google has some very cool n-gram data sets available for download. Alas, the files are quite large. For example, the 3-grams are split into 200 zip files, each weighing in at 440 MB. That’s about 88 GB total.

The 1-grams are much lighter, totaling 2 GB. I was able to reduce this to about 35 MB by throwing away the time information (the original files indicate the year each data point came from). That’s smaller than the original files by a factor of 57.

1-grams in lexicographical order | 1-grams by count | 1-grams by length

The 2-grams, 3-grams, and so on could also be similarly compressed by anyone who can get the files. I’d be happy to help if anyone wants to do this.

November 19, 2011

Kanji with Combinatorial Power

Filed under: General — Peter de Blanc @ 10:38 am

The Japanese writing system includes hiragana, katakana, and kanji. While hiragana and katakana represent sounds, kanji represent meanings. There are a lot of good places to start with learning Kanji. The JLPT N5 kanji list is a set of kanji that appear in the first of 5 standardized tests. Here is a list of the 100 most commonly-used Kanji on the internet.

Kanji may represent words by themselves, or they may be combined into larger units of meaning called compounds. I was interested in finding a set of kanji with a lot of combinatorial power — the ability to represent many distinct compounds without using any kanji not in the set.

For this project I used Jim Breen, jisho.org, Python, and a greedy heuristic. This solution is unlikely to be optimal. I’d like to share a truly optimal solution (with proof) later, but I don’t know how long that will take. Here’s what I’ve got now:

1. counter for days
day
Japan
sun
2. fix
frankness
honesty
repair
straightaway
3. birth
genuine
life
4. center
in
inside
mean
middle
5. person 6. back
behind
later
7. before
in front
8. one
9. counter for years
year
10. month
moon
11. half
middle
odd number
part-
semi-
12. evening
night
13. leader
long
14. interval
space
15. fate
figures
law
number
strength
16. hour
time
17. 1%
chances
degree
duty
know
minute of time
one’s lot
part
rate
segment
shaku/100
share
understand
18. what
19. symbol for kanji repetition 20. book
counter for long cylindrical things
main
origin
present
real
true
21. govern
honorable
manipulate
22. hand
23. big
large
24. expert
family
home
house
performer
professional
25. mountain 26. little
small
27. country 28. three
29. below
descend
down
give
inferior
low
30. above
up
31. one’s station in life
person
somebody
32. be sufficient
counter for pairs of footwear
foot
leg
33. age
change
convert
counter for decades of ages, eras, etc.
fee
generation
period
rate
replace
substitute
34. distinguished
name
noted
reputation
35. matter
object
thing
36. used to indicate possession
(actually a hiragana)
37. 11PM-1AM
child
first sign of Chinese zodiac
sign of the rat
38. female
woman
39. learning
science
study
40. bureau
class
copy
counter for copies of a newspaper or magazine
dept
part
portion
section
41. art
decoration
figures
literature
plan
sentence
style
42. heavens
imperial
sky
43. body
counter for images
object
reality
substance
44. air
atmosphere
mind
mood
spirit
45. alternative
direction
person
46. earth
ground
47. heart
mind
spirit
48. care
class
experience
eye
favor
insight
look
49. two 50. ten
51. color 52. course
district
journey
moral
road-way
street
teachings
53. water 54. gods
mind
soul
55. chief
lord
main thing
master
principal
56. face
features
mask
surface
57. 1040
correct
justice
righteous
58. character
letter
section of a village
word
59. gold 60. harmony
Japan
Japanese style
peace
soften
61. beginning
former time
origin

Here are the compounds (1309 items):

日, 日日, 直, 直直, 日直, 生, 生生, 生日, 中, 中生, 中日, 直中, 日中, 中中, 生中, 人, 人生, 中人, 人人, 人日, 後, 後生, 後日, 人後, 生後, 直後, 後後, 後人, 前, 人前, 生前, 前後, 前人, 前生, 前日, 中前, 直前, 前前, 一, 一直, 一人, 一生, 一日, 一一, 中一, 一人前, 一日中, 一日一日, 一人一人, 後生一生, 年, 後年, 生年, 前年, 中年, 年後, 年生, 年中, 一年, 年年, 一年生, 一年中, 一年前, 月, 一月, 月日, 年月, 月月, 日月, 前月, 月中, 後月, 人月, 年月日, 生年月日, 半, 一半, 後半, 中半, 半生, 前半, 半月, 半日, 半年, 半半, 生半, 後半生, 半人前, 前半生, 一年半, 半年後, 夜, 月夜, 前夜, 日夜, 半夜, 夜前, 夜中, 一夜, 夜半, 夜夜, 夜直, 後夜, 中夜, 一夜中, 夜夜中, 一日一夜, 長, 生長, 長月, 長年, 年長, 半長, 夜長, 長長, 日長, 長夜, 長生, 人長, 直長, 長日月, 間, 一間, 間中, 月間, 人間, 中間, 日間, 年間, 夜間, 間間, 一年間, 数, 数数, 間数, 数人, 数日, 数年, 中数, 年数, 半数, 人数, 日数, 数間, 数日間, 数日後, 数年間, 数年前, 時, 一時, ひととき, 時間, 人時, 日時, 半時, 時時, 時半, 時人, 時日, 時数, 一時間, 長時間, 数時間, 半時間, 分, 一分, 時分, 中分, 年分, 半分, 分数, 夜分, 数分, 分間, 一分半, 時分時, 数分後, 一人分, 数分間, 何, 何月, 何時, 何人, 何日, 何年, 何分, 何何, 何時間, 何年間, 何年生, 何時何時, 何時時分, 何時何分, 々, 数々, 後々, 人々, 年々, 一々, 直々, 月々, 時々, 中々, 長々, 何々, 半々, 日々, 前々, 間々, 夜々, 夜々中, 本, 一本, 本人, 本数, 本日, 本年, 日本, 本々, 本本, 本分, 本月, 本間, 日本間, 日本人, 日本中, 生一本, 日本一, 中日本, 日本時間, 御, 御前, 御中, 御数, 御人, 御御, 御々, 御生, 御年, 御本, 御分, 手, 一手, 後手, 手間, 手数, 手前, 手本, 中手, 手中, 手長, 本手, 人手, 御手, 長手, 手々, 手手, 御手前, 御手数, 御手手, 後手後手, 大, 一大, 御大, 大手, 大人, 大生, 大日, 大半, 大分, 長大, 大本, 大数, 大御, 大前, 大年, 大人数, 家, 家人, 家中, 家長, 後家, 人家, 生家, 大家, 分家, 本家, 一家, 家々, 家家, 長家, 御家人, 山, 山間, 山中, 人山, 前山, 一山, 山々, 山山, 山家, 山手, 山人, 大山, 本山, 御山, 後山, 山月, 大本山, 小, 小山, 小人, 小数, 小生, 大小, 中小, 小手, 小夜, 小一, 小間, 小家, 大中小, 小人数, 小半日, 小半時, 大小前, 小一時間, 国, 一国, 国家, 国手, 国大, 国中, 小国, 大国, 中国, 半国, 本国, 国々, 国国, 国本, 国人, 山国, 生国, 御国, 中間国, 中国人, 本国人, 国家間, 日本国, 三, 三月, 三国, 三時, 三人, 三直, 三日, 三分, 三本, 三年, 三大, 中三, 三家, 御三時, 三国一, 三大国, 三日月, 御三家, 三日間, 三国人, 三年生, 三日三夜, 下, 下家, 下御, 下国, 下山, 下手, 下人, 下半, 時下, 直下, 年下, 下々, 下下, 手下, 月下, 三下, 下前, 下手人, 三下半, 上, 御上, 山上, 上下, 上手, 上人, 上前, 上長, 上半, 直上, 年上, 上々, 上上, 長上, 上家, 上中下, 上御一人, 一上一下, 上手下手, 下手上手, 身, 一身, 身上, 身長, 身分, 生身, 前身, 半身, 分身, 人身, 中身, 長身, 御身, 身中, 小身, 大身, 三身, 直身, 後身, 一身上, 上半身, 下半身, 足, 一足, 下足, 手足, 上足, 人足, 前足, 足下, 後足, 日足, 小足, 大足, 長足, 足長, 生足, 足半, 足中, 三本足, 御御足, 御々足, 代, 一代, 時代, 上代, 前代, 代人, 代数, 大代, 年代, 代々, 代代, 身代, 三代, 後代, 御代, 手代, 足代, 間代, 本代, 月代, 中生代, 大時代, 大御代, 三国時代, 名, 一名, 下名, 家名, 国名, 人名, 数名, 前名, 大名, 本名, 名家, 名手, 名人, 名前, 名月, 名分, 御名, 名山, 名代, 名数, 小名, 何名, 三名山, 日本三名山, 物, 下物, 何物, 御物, 小物, 上物, 人物, 生物, 代物, 大物, 本物, 名物, 一物, 長物, 物日, 物前, 一人物, 下手物, 小間物, 大人物, 時代物, 年代物, 一生物, の, 本の, 何の, 長の, 分の, 大の, 物の, 月の物, 間の手, 身の上, 下の下, 家の人, 山の手, 国の本, 本の間, 本の名, 手の物, 後の月, 小の月, 大の月, 分の一, 日の本, 中の下, 一日の長, 一家の長, 三分の一, 御手の物, 下の名前, 何の何の, 子, 子年, 子分, 年子, 分子, 一子, 御子, 中子, 日子, 長子, 子月, 小子, 間の子, 人の子, 数の子, 中間子, 一人子, 上の子, 子の時, 下の子, 子の日, 分生子, 一の御子, 生物分子, 女, 下女, 三女, 山女, 子女, 女子, 女人, 女生, 女中, 大女, 長女, 一女, 女手, 女御, 女物, 年女, 小女, 日女, 下女中, 女の子, 女の人, 女子大, 上女中, 女の手, 小女子, 女御子, 女子大生, 学, 家学, 学生, 学長, 学年, 学名, 後学, 小学, 数学, 大学, 中学, 夜学, 国学, 本学, 女学生, 小学生, 人間学, 生物学, 代数学, 大学生, 中学生, 日本学, 中学年, 学生時代, 夜間中学, 女子学生, 女子大学, 小中学生, 分子生物学, 部, 一部, 下部, 学部, 後部, 上部, 前部, 大部, 中部, 部下, 部数, 部長, 部分, 本部, 三部, 小部, 足部, 一部分, 学部生, 学部長, 山間部, 大部分, 夜間部, 女子部, 本部長, 小部分, 学生部長, 学部学生, 中部時間, 文, 一文, 国文, 三文, 上文, 前文, 文学, 文月, 文中, 文部, 名文, 人文, 本文, 文身, 文人, 文名, 文物, 長文, 時文, 下文, 文学部, 国文学, 名文家, 日本文学, 人文学部, 天, 一天, 上天, 中天, 天下, 天国, 天子, 天女, 天上, 天生, 天日, 天分, 天文, 半天, 天人, 後天, 天長, 天の, 天部, 本天, 天一, 天の下, 天下一, 天文家, 天文学, 三日天下, 一人天下, 天下国家, 天上天下, 天一天上, 体, 一体, 三体, 女体, 上体, 人体, 生体, 身体, 体中, 体長, 大体, 天体, 物体, 分体, 文体, 本体, 下体, 国体, 体文, 人体学, 身体中, 中間体, 天体学, 生体学, 半数体, 本体部, 生物体, 気, 一気, 気学, 気前, 気体, 気長, 気分, 上気, 人気, 大気, 中気, 天気, 本気, 夜気, 生気, 女気, 山気, 御天気, 一本気, 上天気, 物の気, 大人気, 山の気, 方, 一方, 下方, 何方, 後方, 前方, 大方, 身方, 御方, 三方, 女方, 上方, 方々, 方方, 子方, 後の方, 長方体, 直方体, 御前方, 天文方, 手前の方, 地, 下地, 山地, 生地, 代地, 大地, 地下, 地上, 地代, 地方, 地名, 分地, 天地, 地文, 地学, 地中, 地物, 地気, 本地, 地子, 人地名, 地方時, 天地人, 地の文, 小天地, 地文学, 本地身, 地方名, 半地下, 中国地方, 中部地方, 心, 一心, 下心, 気心, 手心, 小心, 心気, 心身, 身心, 心地, 心中, 中心, 天心, 物心, 本心, 人心, 心天, 女心, 地心, 心学, 御心, 人心地, 中心地, 心の中, 中心部, 中心体, 中心人物, 一家心中, 心身一体, 中心小体, 目, 一目, 下目, 上目, 名目, 目下, 目上, 目前, 目方, 人目, 長目, 文目, 夜目, 後目, 地目, 大目, 目地, 小目, 年目, 本目, 御目, 天目, 何代目, 日の目, 目の前, 天目山, 御目目, 御目々, 名目上, 時間目, 夜の目, 二, 二子, 二月, 二三, 二手, 二足, 二直, 二日, 二部, 二分, 二方, 二目, 二人, 二心, 一二, 中二, 二女, 二本, 二間, 二時, 二の足, 二国間, 二人分, 二人目, 二人前, 二年生, 二日間, 二足三文, 二分の一, 一日二日, 十, 十一, 十月, 十三, 十手, 十数, 十二, 十日, 十分, 十本, 十目, 二十, 十代, 何十, 数十, 三十, 十年, 十地, 十方, 十夜, 三十日, 十一月, 十三夜, 十二月, 十二時, 十二分, 二十代, 二十日, 十二直, 二十年, 数十年, 十日夜, 十年一日, 十分の一, 十月十日, 色, 気色, 月色, 三色, 色気, 色女, 色目, 天色, 二色, 物色, 夜色, 一色, 色々, 色色, 地色, 女色, 間色, 目色, 色文, 色物, 本色, 生色, 体色, 色身, 色数, 色子, 地方色, 中間色, 目の色, 時代色, 十人十色, 道, 一道, 下道, 気道, 国道, 山道, 小道, 人道, 地道, 中道, 二道, 半道, 本道, 夜道, 天道, 間道, 道中, 道心, 道学, 道家, 道道, 道々, 大道, 色道, 道長, 一本道, 地下道, 何方道, 何の道, 人間道, 上気道, 中山道, 水, 一水, 下水, 山水, 小水, 上水, 水気, 水子, 水上, 水色, 水生, 水中, 水道, 水手, 水分, 大水, 天水, 生水, 手水, 水心, 水物, 水部, 水目, 三水, 二水, 水文, 名水, 下水道, 上水道, 中水道, 地下水, 水道水, 水文学, 水上下, 水の道, 上下水道, 水生生物, 水道の水, 神, 女神, 神身, 身神, 神学, 神代, 神道, 半神, 天神, 神々, 神神, 神子, 神月, 水神, 神人, 神気, 神色, 神体, 神国, 神前, 心神, 色神, 大神, 月神, 神名, 中神, 年神, 御神体, 大御神, 直日神, 天一神, 女神子, 大年神, 神の子, 山の神, 国家神道, 主, 家主, 主家, 主上, 主色, 主体, 主長, 主日, 主部, 主物, 主文, 神主, 水主, 地主, 天主, 道主, 主人, 主神, 国主, 名主, 女主, 御主, 御主人, 主の日, 女主人, 大地主, 女地主, 一家の主, 主の御名, 大国主神, 面, 一面, 月面, 三面, 小面, 上面, 人面, 水面, 前面, 体面, 地面, 直面, 半面, 部面, 文面, 方面, 面子, 面前, 面目, 面々, 面面, 面色, 面体, 面長, 面部, 生面, 天面, 下面, 水面下, 一生面, 水の面, 正, 正月, 正子, 正色, 正数, 正体, 正大, 正中, 正直, 正物, 正文, 正方, 正本, 正面, 正目, 大正, 中正, 正気, 正道, 正々, 正正, 方正, 正長, 文正, 天正, 正身, 小正月, 正の数, 中正面, 大正時代, 大正の代, 目の正月, 二十日正月, 字, 一字, 国字, 字体, 十字, 数字, 正字, 大字, 二字, 文字, 本字, 名字, 字面, 字数, 小字, 字間, 一文字, 十文字, 小文字, 大文字, 御の字, 人文字, 女文字, 大の字, 二文字, 文字間, 神代文字, 一字一字, 御目文字, 中間文字, 名前文字, 小文字名前文字, 大文字名前文字, 金, 御金, 下金, 金物, 金目, 後金, 小金, 正金, 代金, 大金, 天金, 年金, 半金, 金色, 前金, 金子, 金気, 地金, 金山, 金字, 金主, 手金, 月金, 金文, 金神, 水金, 一時金, 月水金, 身代金, 一分金, 金文字, 人の金, 身の代金, 金目の物, 和, 中和, 日和, 和人, 和金, 和上, 和文, 和本, 大和, 和名, 和字, 和気, 和物, 長和, 正和, 文和, 天和, 水和, 和主, 直和, 和国, 和方, 和子, 大和心, 水和物, 大和文, 和文体, 大和時代, 大和の国, 一水和物, 大和国家, 元, 一元, 家元, 元気, 元三, 元手, 元日, 元年, 元方, 元本, 手元, 身元, 地元, 中元, 元金, 足元, 元々, 元元, 国元, 山元, 道元, 目元, 本元, 二元, 天元, 長元, 正元, 元中, 元和, 元文, 元正, 御中元, 三元色, 御手元, 本家本元.

October 10, 2011

Directions for this Blog

Filed under: General — Peter de Blanc @ 8:36 am

It’s been nearly a year since my last post here. I’d like to get back into blogging, so I’ll start by listing various things I might blog. Here’s your chance to affect what I write about.

The list:

  1. My new organization, It Builds Itself.
  2. Books, articles, and papers I’m reading.
  3. Programs, programming, and programming languages.
  4. Go, programs that play Go, math about Go, and statistics about Go.
  5. Details, possibly excessive, about my personal life.
  6. The singularity and the future of humanity and of the universe (topics about which I know very little).
  7. Mathematics that is already widely understood, but which may be new to some folks.
  8. Imaginary places, people, and things.

Since this ended up being a pretty short list, I’ll be accepting further suggestions as well.

October 19, 2010

Space and Games is on Twitter

Filed under: General — Peter de Blanc @ 4:20 pm

I haven’t been posting on this blog much lately, but I have been tweeting. If you want to read more of my stuff, follow @spaceandgames.

March 14, 2010

The War Club vs. the Ant-Men, part 1

Filed under: General — Peter de Blanc @ 11:19 pm

Paron had been at the academy for far too long. He had switched from geometry to biology, and finally to game theory. When at last he finished his thesis, it was a cause for celebration. The party was far from grand; more than twenty people were packed into Paron’s meager, candle-lit apartment. Like organisms, conversations competed with their kin for limited attention and limited air while sleep-deprived gamers competed to dominate virtual markets. I was in my element.

“The strange thing about Ant War,” I mused, “is the player. We’re willing to anthropomorphize ants to the point of substituting our own decisions for theirs in the game, but ant behavior is completely inhuman.”

“It’s not like normal human behavior,” said my opponent, Nik, “but unusual situations can produce unusual behaviors. We’re fighting a virtual war; humans could also fight a real one.”

This comment drew Paron’s attention. “More unusual than you might think. A war requires extreme cooperation. The ants in a colony are all sisters, and they share 3/4 of their genes because their father is basically a glorified sperm cell. To get humans to cooperate like ants, they’d have to be born of incest.”

“You don’t need common end goals to cooperate,” countered Nik, “only common proximate goals. Not to mention the fact that evolution formed our goal systems imperfectly; we may agree on values aside from inclusive fitness.”

“But a war requires two coalitions,” said Paron. “You’d need everyone in your coalition to share proximate goals – and everyone in the opposing coalition to share an opposing proximate goal. And if both coalitions are composed of humans? — what could ever cause that, apart from inclusive fitness?”

I moved my ants, then re-entered the conversation. “What about competing protocols? Maybe one coalition follows one set of rules, and the other coalition follows different rules. Economic productivity would be boosted if everyone followed the same rules, but whichever coalition is forced to convert has to pay a cost.”

“But then you’d just bid on it,” said Paron.

“Why don’t ants bid on land?” asked Nik.

Paron said, “Ants don’t engage in inter-colony trade. They’re not smart enough to trade, so they war instead.”

“Let’s go back to end goals,” said Nik. “I think there are goals that all humans share; we all value love, life, and laughter, and not just for ourselves and our kin. One coalition could be human. The opposing coalition could be inhuman.”

“Like what? Giant ants?” I asked.

Nik said, “Or the broad-shouldered people across the sea.”

January 19, 2010

RPS Equilibrium Conundrum

Filed under: General — Peter de Blanc @ 10:05 pm

Clearly, it’s absurd that paper beats rock, but if rock beat paper then the game would become pointless.

Suppose we changed the rules such that paper only scores 1/2 point against rock. A full victory (rock against scissors or scissors against paper) scores 1 point, and a loss scores -1 point. Draws score 0 points. What mixed strategy is best in this game?

I found this equilibrium: p(rock) = p(paper) = 2/5, and p(scissors) = 1/5. If the opponent plays this strategy, then anything we do has an expected utility of 0. If both players use this strategy, then neither player has an incentive to change, so it’s an equilibrium.

This result seems strange to me. The rule change makes paper worse, and yet in the resulting equilibrium, we increase the probability of throwing paper. Who wants to explain this?

December 28, 2009

Germs, Selection, and Disease

Filed under: General — Peter de Blanc @ 10:35 am

The germ that infects you was selected for its ability to spread across hosts, but the germ population in your body is being selected for its ability to spread within your body. The latter is more destructive than the former, so the germ population in your body becomes more destructive over time. Thus for any infectious disease, we should expect the period of maximal transmissibility to precede the period of maximal suffering.

November 4, 2009

Go proverbs: “A rich man should not pick quarrels.”

Filed under: General, Go — Peter de Blanc @ 1:19 pm

Go players have hundreds of proverbs — pithy sentences that convey important heuristics. It is not enough to simply read proverbs; you must study them at length to unfold them into procedural knowledge.

Most proverbs are particular to Go (e.g. six die but eight live), but some generalize to other adversarial situations, and a few proverbs contain important lessons about rationality.

One of my favorite proverbs states that a rich man should not pick quarrels. Go, in its most common formulations, is a game of satisficing. The player with more points wins the game, and winning is enough; there is no extra reward for winning by a large margin. The proverb says that if you are currently winning (i.e. you are a rich man), then you should not do things (such as picking quarrels) that make the outcome more random. By decreasing the variance in the probability distribution for your final score, you increase the probability that you will hold onto enough points to win. Anything that makes the game simpler and more predictable is good for you.

We can see this in Chess (the winning player should seek to trade pieces) and in epee fencing (the winning player should seek double-touches).

If, on the other hand, you are a poor man, then you should pick quarrels. There’s a good example of this in Indiana Jones and the Temple of Doom. In one scene, Indy is in the middle of a rope bridge, and swordsmen are approaching from either side, so Indy cuts the bridge.

If you are winning, simplify. If you are losing, complexify.

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