Counting Awari Positions…
Previously I wrote a simple program to compute the number of positions that are legal in Checkers. I thought I might perform the same analysis for the game of Awari. The simplest approximation that you can get is by counting the number of ways that n undistinguished stones can be distributed in m distinguished pits. A few minutes scratching (or searching the web) will show you that n undistinguished balls can be distributed in m distinguished pits in comb(n+m-1, m-1) ways. Awari begins with 48 stones and 12 pits, so if we sum all the totals, we’ll get the total number of positions. Well, it is an overestimate. First of all, there are no legal positions in Awari with 47 stones, since the first capture either captures 2 or 3 stones. And, there are many positions in here which cannot occur in the game. For instance, in the real game, the non-moving player can never have stones in every pit (except the first move, of course) because he would have previously had to empty one pit to sow it, and that pit is always skipped when you sow. And these totals don’t include any kind of “reachability” analysis: many nodes can’t be achieved in play.
According to Romein and Bal’s work, there are 889,063,398,406 positions which are reachable in Awari. Writing up my naive totaller gives the following totals:
-------------------- 1 12 2 78 3 364 4 1,365 5 4,368 6 12,376 7 31,824 8 75,582 9 167,960 10 352,716 11 705,432 12 1,352,078 13 2,496,144 14 4,457,400 15 7,726,160 16 13,037,895 17 21,474,180 18 34,597,290 19 54,627,300 20 84,672,315 21 129,024,480 22 193,536,720 23 286,097,760 24 417,225,900 25 600,805,296 26 854,992,152 27 1,203,322,288 28 1,676,056,044 29 2,311,801,440 30 3,159,461,968 31 4,280,561,376 32 5,752,004,349 33 7,669,339,132 34 10,150,595,910 35 13,340,783,196 36 17,417,133,617 37 22,595,200,368 38 29,135,916,264 39 37,353,738,800 40 47,626,016,970 41 60,403,728,840 42 76,223,753,060 43 95,722,852,680 44 119,653,565,850 45 148,902,215,280 46 184,509,266,760 48 279,871,768,995 -------------------- 1,171,666,558,334
which indicates that approximately 1/4 of all the positions enumerated here are illegal or unreachable. Still, the ease of indexing using this naive idea may compensate for doing anything more compact but more confusing.
I suspect the world would be better if that percentage were even greater.
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