## Friday, December 15, 2017

### [agobrown] Longest games of chomp

What Chomp starting positions offer the longest games, perhaps the most possibilities for interesting games?  Among rectangular starting positions, good starting positions are 13x12, 12x11, 10x9, 9x8, 11x6, 7x6, 8x5, 6x5, 5x4.  Missing from the pattern of (N)x(N-1) are 11x10 and 8x7.  (Chomp is weird in how there aren't simple patterns.  It might be a good candidate for machine learning.)

We assumed 3 types of positions in Chomp are instantly known lost (P positions):

1. L-shaped positions with both arms of the L having unit width and same lengths
2. 2-row positions of the form [a,a-1]
3. 3-row positions of the form [a,a-2,2]

The 3-row [a,a-2,2] class of positions is noted in Proposition 2 of "Three-Rowed Chomp" by Doron Zeilberger.  The winning strategy from such a position is as follows:

The base case is [4,2,2] (which looks kind of like a pistol).  If the opponent moves to [3,2,2], then respond moving to [3,2] and follow the 2-row strategy (or move to [3,1,1] and L-shaped strategy).  If [2,2,2] then 2-row strategy vertically.  If [4,1,1] then [3,1,1] and L-shaped strategy.  If [4,2,1] then [2,2,1] and 2-row strategy vertically.  If [4,2] then 2-row strategy.

For larger 3-row positions [a,a-2,2], if the opponent moves in the first 2 rows, leaving at least 4 in the first row and at least 2 in the second row, then restore the position to the shape [b,b-2,2].  If [3,3,2] then [3,1,1] and L-shaped strategy.  If [a,1,1] then [3,1,1] and L-shaped strategy.  If the opponent moves on the third row to [a,a-2,1] then [2,2,1] and follow the 2-row strategy vertically.  If [a,a-2], then 2-row strategy.

Here is complete output of all positions within board size 13x13 and Haskell source code.  A selection of some positions and their game values are also given below.  Computing board size 12 required 8.5 GB of RAM on a machine with 16 GB of RAM.  (Haskell programs tend to use a lot of memory unless one puts effort into conserving memory, which we did not do.)

For computing board size 13, we allowed swapping to virtual memory on SSD on a machine with 8 GB of physical RAM.  The output of /usr/bin/time was:

5751.60user 86808.57system 39:48:33elapsed 64%CPU (0avgtext+0avgdata 7192640maxresident)k
10410518744inputs+8outputs (184956202major+316491058minor)pagefaults 0swaps

This suggests a slowdown factor of about 25 for using virtual memory on SSD compared to RAM for this program which made heavy use of Data.Map.  Polling "ps xu" saw a maximum virtual memory usage of 39 GB.  For the output of the board size 13 at the link above, we omitted saving the "Win_in 1" positions to save disk space.

There are only 3 "Lose in 2" positions: [6,3,3]; [5,5,3]; and [5,2,1,1].  Memorize them to get an edge against opponents.  One could also memorize the 7 "Lose in 4" positions, 14 "Lose in 6", 26 "Lose in 8"...

There seem to be some more patterns that lose: [odd,2,1,1,1,...]; [even,3,1,1,1,...]; [even,2,2,2,1,1,1,...]; [even,2,2,1,1,1,...]; [odd,4,1,1,1,...].  These deserve future investigation.  Andries Brouwer's web site suggests that losing families of positions exist in 3-row chomp for [a+11,a+7,5]; [?,?,7]; [?,?,9]; [?,?,11]; [?,?,14] (not 13, once again breaking what seemed to be a simple pattern of odd third rows).  It still needs to be explicitly articulated how to win after giving your opponent these losing positions.  Work by Steven Byrnes suggests the game values of all 3-row Chomp positions can be rapidly computed, though probably not by a human in his or her head.  Future versions of the code should bound not by board size but number of pieces, to investigate thin positions and roughly L-shaped positions.

(Position [13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 12, 5], Win_in 103)
(Position [13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 5], Win_in 103)
(Position [13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13], Win_in 101)
(Position [12, 12, 12, 12, 12, 12, 12, 12, 12, 10, 7], Lose_in 86)
(Position [12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12], Win_in 79)
(Position [11, 11, 11, 10, 10, 10, 10, 10, 2], Win_in 57)
(Position [11, 11, 11, 10, 10, 10, 10, 9, 2], Win_in 57)
(Position [11, 11, 11, 11, 11, 9, 9, 7, 1, 1], Win_in 57)
(Position [11, 11, 11, 11, 11, 9, 9, 9, 1, 1], Win_in 57)
(Position [11, 11, 11, 11, 11, 11], Win_in 43)
(Position [11, 11, 11, 11, 11, 11, 11], Win_in 41)
(Position [11, 11, 11, 11, 11, 11, 11, 11], Win_in 39)
(Position [11, 11, 11, 11, 11, 11, 11, 11, 11], Win_in 37)
(Position [11, 11, 11, 11, 11], Win_in 35)
(Position [11, 11, 11, 11, 11, 11, 11, 11, 11, 11], Win_in 21)
(Position [10, 10, 10, 10, 10, 10, 10, 10, 4], Lose_in 56)
(Position [10, 10, 10, 10, 10, 10, 10, 10, 10], Win_in 55)
(Position [9, 9, 9, 9, 9, 9, 9, 9], Win_in 41)
(Position [8, 8, 8, 8, 8], Win_in 23)
(Position [8, 8, 8, 8, 8, 8], Win_in 15)
(Position [8, 8, 8, 8, 8, 8, 8], Win_in 13)
(Position [7, 7, 7, 7, 7, 7], Win_in 21)
(Position [6, 6, 6, 6, 2], Win_in 13)
(Position [6, 6, 6, 6, 6], Win_in 9)
(Position [5, 5, 5, 5], Win_in 5)
(Position [4, 4, 4, 4], Win_in 1)
(Position [4, 4, 4], Win_in 1)
(Position [4, 4], Win_in 1)
(Position [4], Win_in 1)

(Position [5, 2, 1, 1], Lose_in 2)
(Position [5, 5, 3], Lose_in 2)
(Position [6, 3, 3], Lose_in 2)

(Position [5, 3, 3, 2], Lose_in 4)
(Position [5, 5, 2, 2], Lose_in 4)
(Position [6, 2, 2, 1, 1], Lose_in 4)
(Position [6, 2, 2, 2], Lose_in 4)
(Position [6, 3, 1, 1, 1], Lose_in 4)
(Position [7, 2, 1, 1, 1, 1], Lose_in 4)
(Position [7, 4, 3], Lose_in 4)

(Position [6, 4, 3, 3, 2], Lose_in 6)
(Position [7, 2, 2, 2, 2], Lose_in 6)
(Position [7, 3, 2, 1, 1, 1], Lose_in 6)
(Position [7, 3, 2, 2], Lose_in 6)
(Position [7, 3, 3, 1, 1], Lose_in 6)
(Position [7, 3, 3, 2, 1, 1], Lose_in 6)
(Position [7, 4, 1, 1, 1], Lose_in 6)
(Position [7, 5, 3, 2], Lose_in 6)
(Position [7, 7, 4], Lose_in 6)
(Position [8, 2, 2, 1, 1, 1, 1], Lose_in 6)
(Position [8, 2, 2, 2, 1, 1], Lose_in 6)
(Position [8, 3, 1, 1, 1, 1, 1], Lose_in 6)
(Position [8, 4, 4], Lose_in 6)
(Position [9, 2, 1, 1, 1, 1, 1, 1], Lose_in 6)

(Position [6, 4, 4, 3, 3], Lose_in 8)
(Position [6, 6, 3, 3, 3], Lose_in 8)
(Position [6, 6, 4, 3, 2], Lose_in 8)
(Position [7, 3, 3, 3, 2, 2], Lose_in 8)
(Position [7, 4, 2, 2, 2, 2], Lose_in 8)
(Position [7, 4, 4, 2], Lose_in 8)
(Position [7, 5, 3, 3, 1, 1], Lose_in 8)
(Position [7, 7, 3, 3], Lose_in 8)
(Position [8, 3, 2, 2, 2], Lose_in 8)
(Position [8, 3, 3, 3], Lose_in 8)
(Position [8, 4, 2, 1, 1, 1], Lose_in 8)
(Position [8, 4, 2, 2], Lose_in 8)
(Position [8, 5, 1, 1, 1], Lose_in 8)
(Position [8, 5, 4, 2], Lose_in 8)
(Position [9, 2, 2, 2, 2, 1, 1], Lose_in 8)
(Position [9, 2, 2, 2, 2, 2], Lose_in 8)
(Position [9, 3, 2, 1, 1, 1, 1, 1], Lose_in 8)
(Position [9, 3, 2, 2, 1, 1, 1], Lose_in 8)
(Position [9, 4, 1, 1, 1, 1, 1], Lose_in 8)
(Position [9, 4, 4, 1, 1], Lose_in 8)
(Position [9, 5, 3, 1, 1, 1, 1], Lose_in 8)
(Position [9, 5, 4], Lose_in 8)
(Position [10, 2, 2, 1, 1, 1, 1, 1, 1], Lose_in 8)
(Position [10, 2, 2, 2, 1, 1, 1, 1], Lose_in 8)
(Position [10, 3, 1, 1, 1, 1, 1, 1, 1], Lose_in 8)
(Position [11, 2, 1, 1, 1, 1, 1, 1, 1, 1], Lose_in 8)