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# John Horton Conway

Wikipedia

#### Jon Diamond

"Theory of sums of partizan games"

Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness
by Donald E. Knuth.     (video, book)

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# Partizan games

Nim is an impartial game, because both players have the same 'moves'.
Go is a partizan game, because it has different 'moves' for each player.

• First player loses
• Left player wins
• Right player wins
• Second player loses
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# Go end-game

(No life-and-death situations left)
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``` (White to play and win)

Experiments in Computer Go Endgames
by Martin Müller and Ralph Gasser
(PDF)

Generalized Thermography: Algorithms, Implementation, and Application to Go Endgames
by Martin Müller, Elwyn Berlekamp, and Bill Spight
(PDF)

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# Combining 'games' ```

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Miai counting
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# Game definition

{ left-moves | right-moves }

Empty game: { | } = 0
Left wins: { {|} | } = 1
Right wins: { | {|} } = -1
First wins: { {|} | {|} } = *
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If x = {L|R} then:
xL a typical member of L
xR a typical member of R

x + y = {xL + y, x + yL | xR + y, x + yR}

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# Surreal numbers

{ | } = 0
{ 0 | } = 1
{ | 0 } = -1
{ 1 | } = 2
{ 2 | } = 3
{ 0 | 1 } = 1/2
{ 1 | 2 } = 3/2
{ 0 | 1/2 } = 1/4

{ 0, 1, 2, 3, … | } = ω
{ 0 | 1, 1/2, 1/4, 1/8, … } = ε

π = {3, 25/8, 201/64, … | …, 51/16, 13/4, 7/2, 4}

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# Are they numbers?

x ≤ y iff (no yR ≤ x and y ≤ no xL)
-x = { -xR | -xL }
xy = { xLy + xyL - xLyL, xRy + xyR - xRyR
| xLy + xyR - xLyR, xRy + xyL - xRyL }
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# Calculator

Input:

Output:

Trace:

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