Perft

Perft is a heuristic for testing a game engine's valid move generator, developed by chess programmers, by calculating how many possible board positions exist a certain number of turns in the future.

To calculate the "perft result" one walks (plays) the tree of every possible legal move to a certain ply (turn) depth and counts the number of leaf nodes (final positions). The results can then be compared against a pre-computed table to validate that the engine is generating the correct number of legal moves at each board position.

Initial Position Example

By definition, perft(0) = 1, since if you don't play any moves, you only have the same board position you started with.

Now, following the rules of Hive as implemented by Mzinga (see notes below), for a new game with no expansion pieces at the initial position, White has four options (place either a Beetle, a Grasshopper, a Spider, or an Ant).

Therefore with four possible board positions after one turn, perft(1) = 4.

After that first turn, Black can also respond with either a Beetle, a Grasshopper, a Spider, or an Ant, and can place their bug on any of the six available sides of White's first bug. So Black has (6 x 4) = 24 possible responses to White's first play. If we multiply the number of plays White could have played times the number of play Black could respond with, you get (4 x 24) = 96 possible board positions after two turns. Therefore perft(2) = 96.

After the first two turns, as Queens can come into play and pieces can not only be placed but also moved, calculating perft by hand becomes much more difficult.

Perft Results (Initial Position)

All of these numbers were calculated using MzingaPerft v0.12.9 following the notes below.

Depth	Base	Base+M	Base+L	Base+P
0	1	1	1	1
1	4	5	5	5
2	96	150	150	150
3	1,440	2,610	2,610	2,610
4	21,600	45,414	45,414	45,414
5	516,240	1,252,800	1,252,800	1,255,932
6	12,219,480	34,233,432	34,233,672	34,395,984
7	181,641,900	527,164,524	529,630,188	532,753,872
8	2,657,392,800	8,000,790,798	8,072,006,754	8,134,286,034
9	99,376,027,356	341,738,424,810	345,842,146,422	346,017,781,674

Depth	Base+ML	Base+MP	Base+LP	Base+MLP
0	1	1	1	1
1	6	6	6	7
2	216	216	216	294
3	4,320	4,320	4,320	6,678
4	86,400	86,400	86,400	151,686
5	2,725,920	2,730,888	2,730,240	5,427,108
6	85,201,200	85,492,248	85,457,136	192,353,904
7	1,357,078,404	1,363,837,116	1,366,372,440	3,151,035,948
8	21,314,716,308	21,467,457,180	21,547,245,672	50,945,151,390
9	1,036,739,513,856	1,038,414,757,560	1,043,459,509,212	2,784,830,280,258

Note: Numbers in italics have not been independently verified.

Performance Comparisons

Another important use for perft is to determine how quickly an engine can calculate the number of valid moves at any position. By timing how long it takes to calculate a perft result, one can do engine-to-engine performance comparisons, which is usually reported in thousands of nodes visited per second (KN/s), i.e. # of nodes / the time in milliseconds.

Mzinga Perft Performance (Initial Position)

Results from MzingaPerft v0.12.9 on an AMD Ryzen 7 1700X (8 cores @ 3.40GHz) with 32GB RAM.

Depth	Base	Base+M	Base+L	Base+P
0	00:00:00.006 (0.2 KN/s)	00:00:00.006 (0.2 KN/s)	00:00:00.006 (0.2 KN/s)	00:00:00.006 (0.2 KN/s)
1	00:00:00.003 (1.3 KN/s)	00:00:00.003 (1.7 KN/s)	00:00:00.003 (1.7 KN/s)	00:00:00.003 (1.7 KN/s)
2	00:00:00.003 (32.0 KN/s)	00:00:00.003 (50.0 KN/s)	00:00:00.003 (50.0 KN/s)	00:00:00.003 (50.0 KN/s)
3	00:00:00.004 (360.0 KN/s)	00:00:00.006 (435.0 KN/s)	00:00:00.004 (652.5 KN/s)	00:00:00.004 (652.5 KN/s)
4	00:00:00.003 (7,200.0 KN/s)	00:00:00.008 (5,676.8 KN/s)	00:00:00.005 (9,082.8 KN/s)	00:00:00.005 (9,082.8 KN/s)
5	00:00:00.049 (10,535.5 KN/s)	00:00:00.103 (12,163.1 KN/s)	00:00:00.103 (12,163.1 KN/s)	00:00:00.111 (11,314.7 KN/s)
6	00:00:00.802 (15,236.3 KN/s)	00:00:01.893 (18,084.2 KN/s)	00:00:01.935 (17,691.8 KN/s)	00:00:02.009 (17,120.9 KN/s)
7	00:00:23.859 (7,613.1 KN/s)	00:01:02.989 (8,369.2 KN/s)	00:01:08.755 (7,703.2 KN/s)	00:01:08.723 (7,752.2 KN/s)
8	00:06:18.051 (7,029.2 KN/s)	00:18:20.308 (7,271.4 KN/s)	00:18:57.761 (7,094.6 KN/s)	00:20:20.554 (6,664.4 KN/s)
9	03:11:51.473 (8,632.8 KN/s)	10:19:12.401 (9,198.3 KN/s)	10:50:21.048 (8,863.0 KN/s)	11:05:24.268 (8,666.9 KN/s)

Depth	Base+ML	Base+MP	Base+LP	Base+MLP
0	00:00:00.006 (0.2 KN/s)	00:00:00.006 (0.2 KN/s)	00:00:00.006 (0.2 KN/s)	00:00:00.006 (0.2 KN/s)
1	00:00:00.003 (2.0 KN/s)	00:00:00.003 (2.0 KN/s)	00:00:00.003 (2.0 KN/s)	00:00:00.003 (2.3 KN/s)
2	00:00:00.004 (54.0 KN/s)	00:00:00.004 (54.0 KN/s)	00:00:00.004 (54.0 KN/s)	00:00:00.004 (73.5 KN/s)
3	00:00:00.004 (1,080.0 KN/s)	00:00:00.004 (1,080.0 KN/s)	00:00:00.004 (1,080.0 KN/s)	00:00:00.004 (1,669.5 KN/s)
4	00:00:00.009 (9,600.0 KN/s)	00:00:00.010 (8,640.0 KN/s)	00:00:00.009 (9,600.0 KN/s)	00:00:00.014 (10,834.7 KN/s)
5	00:00:00.206 (13,232.6 KN/s)	00:00:00.212 (12,881.5 KN/s)	00:00:00.196 (13,929.8 KN/s)	00:00:00.325 (16,698.8 KN/s)
6	00:00:04.349 (19,591.0 KN/s)	00:00:04.535 (18,851.7 KN/s)	00:00:04.489 (19,037.0 KN/s)	00:00:09.716 (19,797.6 KN/s)
7	00:02:40.616 (8,449.2 KN/s)	00:02:49.430 (8,049.6 KN/s)	00:02:54.094 (7,848.5 KN/s)	00:06:25.217 (8,179.9 KN/s)
8	00:48:21.657 (7,345.7 KN/s)	00:51:03.763 (7,006.9 KN/s)	00:51:32.991 (6,966.5 KN/s)	01:57:58.621 (7,197.0 KN/s)
9	30:58:20.189 (9,298.1 KN/s)	32:36:48.907 (8,844.4 KN/s)	32:34:13.669 (8,899.2 KN/s)	85:38:11.591 (9,033.1 KN/s)

Mzinga Perft Notes

Mzinga follows these rules when generating moves:

The Queen Bee cannot be played on a player's first turn.
If a player has multiple bugs of the same type in their hand, then only one is used to calculate the number of valid placements. I.e. if you have two ants in your hand, and five open positions to place a bug, you have five valid moves, not ten.
A player can only Pass if there are no other legal moves, and then that Pass counts as one move for the purposes of calculating perft.
When the game is over (one or both Queen Bees are surrounded), then the move generator should return 0 moves.