Massacre Proof Games

A game of chess is called a "massacre" if almost every move is a capture. When only the final diagram and the number of moves are given, deducing the game is a chess problem called a "massacre proof game".

Let x be the number of non-captures. The case x=2 is uninteresting because at most 4 halfmoves can be played. The table below contains information about the cases x=3,4,5. The number of pieces left on the board is of course simply 32+x-ply.

Toward the end of the table you will find some links to the corresponding chess problems. The fields inside each list of problems are:

Number of distinct chess positions with exactly x non-captures
  x=3 x=4 x=5
ply 0 0 0 0
ply 1 0 0 0
ply 2 0 0 0
ply 3 5330 0 0
ply 4 1579 70476 0
ply 5 1848 49758 770912
ply 6 1948 79313 1002112
ply 7 1572 112297 2324252
ply 8 1859 166847 4079067
ply 9 2243 235082 7456474
ply 10 3718 351119 13322408
ply 11 5749 581588 23181085
ply 12 9463 967742 40867572
ply 13 14123 1589381 71471685
ply 14 20787 2530833 123150384
ply 15 29323 3808155 202209472
ply 16 43978 5739797 324453946
ply 17 64134 8420818 500029995
ply 18 90659 12289682 760082705
ply 19 121380 16981034 1099990237
ply 20 143938 22302528 1543092047
ply 21 160061 27038573 2012263317
ply 22 149259 29971375 2469906062
ply 23 133316 30457039 2761290606
ply 24 98315 27518641 2829524382
ply 25 72997 22702677 2599742382
ply 26 41924 16157621 2124497143
ply 27 23453 10355382 1522979675
ply 28 9721 5498926 928366242
ply 29 3902 2501639 473870909
ply 30 890 873024 189297438
ply 31 161 229895 56139054
ply 32 17 38842 10835051
ply 33 2 3516 1141076
ply 34 0 90 46697
ply 35 0 0 416
ply 36 0 0 0

Plot of the table entries

To show the similarities, I'm scaling the x=3,4 graphs to the level of the x=5 graph by using some fudge factors.

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