Endgame statistics with fantasy pieces

Dave McCooey has carried out an extensive computer analysis of endgames with two kings, and two pieces, on an 8 by 8 chessboard, that involve at least one fantasy chess piece, taken from a set. Here, you find what he found.

Also, I asked Dave about the endgames with two kings and one fantasy piece. His answer to this question can be found at the end of this file.

See also:

Here are the endgame summaries for all possible 2-piece endgames on an 8x8 square chessboard involving at least one fantasy piece from the following set:
Amazon (symbol = A)
Moves like a Queen or a Knight
Chancellor (symbol = C)
Moves like a Rook or a Knight
Pegasus (symbol = G)
Moves like a Bishop or a Knight
Knightrider (symbol = I)
Moves like a Knight except that it may continue moving in a straight line through unoccupied squares, each a Knight's move from the last and in the same direction away from the starting square.
Camel (symbol = L)
Moves to any of the 8 squares that are at the opposite corner of a 2-by-4 rectangle from the starting square. It hops from square to square like a Knight. This piece is restricted to a single color (like a Bishop is).
Zebra (symbol = Z)
Moves to any of the 8 squares that are at the opposite corner of a 3-by-4 rectangle from the starting square. It hops from square to square like a Knight. This piece always moves to a square of the opposite color (like a Knight does).
Wazir (symbol = W)
Moves like a Rook, but only one square at a time.
Ferz (symbol = F)
Moves like a Bishop, but only one square at a time.
Whenever an endgame involves two pieces that are restricted to a single color (e.g. Bishop, Camel, Ferz), then two summaries are reported - one for the case where the pieces are on the same color, indicated by (s), and one for the case when they are not, indicated by (d).

Definitions used in the statistics:

Unique Legal Positions:
Each legal position is one of either 4 or 8 equivalent positions due to rotations and/or reflections. If all pieces are on a long diagonal, then the number is 4. Otherwise it is 8. The Unique Legal Positions considers a set of equivalent positions as a single position.
WHITE win:
WHITE can force a win.
BLACK win:
BLACK can force a win.
Simple draw:
The weaker side can force a simplification of the position (i.e. by capture), resulting in a simplified position which, of course, must be a draw.
Perpetual check:
The weaker side cannot force a simplification but can nevertheless force a sequence of positions in which the stronger side cannot escape being checked without either losing or allowing simplification.
Fortress draw:
The weaker side can force neither a simplification nor a perpetual check but can nevertheless hold the draw.
Longest win:
The maximum number of half-moves that it can require to force a win.
Here are some observations:
  1. The Pegasus is better than the Rook, although the KGvKR endgame is a draw in general.
  2. The Pegasus is closer in strength to the Rook than to the Chancellor.
  3. The Queen does not beat the Pegasus in general.
  4. The KAvKC endgame has 23 fortress draws. These positions are "reverse perpetual checks" where the strong side (i.e. Amazon) must keep checking the weak side in order to prevent the weak side from either forcing a simplification or getting its own perpetual check! Here are three examples:
    1. WHITE: Kd6, Aa7 BLACK: Kg6, Ch7 (WHITE to move)
    2. WHITE: Kc8, Ab8 BLACK: Kc2, Cd5 (WHITE to move)
    3. WHITE: Kc8, Ah2 BLACK: Kb3, Ce4 (WHITE to move)
  5. A Rook does not beat a Camel in general, but it wins considerably more often than it does against a Bishop or Knight. The longest win can take 35 moves (70 halfmoves).
  6. A Rook beats a Zebra.
  7. A Zebra seems to be almost universally weaker than a Camel, but not by much, of course. This is somewhat unexpected, because the Zebra can reach all 64 squares and the Camel cannot.
  8. A Rook beats a Ferz, which is expected.
  9. A Rook beats a Wazir, which is expected. This endgame contains two sets of fortress draws illustrated by the following positions:
  10. A Pegasus beats a Knightrider, and it can take as long as 57 moves (113 halfmoves) to win.
  11. Two Knightriders can checkmate a lone King.
  12. A Knightrider and any piece from the set {N,L,Z,W,F} can checkmate a lone King.
  13. A Bishop and any piece from the set {L,Z,W,F} can checkmate a lone King, as long as the Camel or Ferz (in such a case) is on a color different than the Bishop's.
  14. The KIFvK endgame has 526 fortress draws, i.e. positions where the lone King can keep the pieces occupied forever. All of these positions have the Knightrider in the corner or one rook-move next to the corner while the Ferz is stuck being protected by the Knightrider. The Black King keeps threatening the Ferz, and the White King is too far away to help.
  15. The KBLvK(d) endgame has 15 fortress draws, i.e. positions where the King can keep the pieces occupied forever. They are divided into two sets of positions:
  16. The KBWvK endgame has exactly 2 (unique) fortress drawn positions and here they are:
    1. WHITE: Kd5, Bc8, Wa8 BLACK: Kc7 (WHITE to move)
    2. WHITE: Kd5, Ba6, Wa8 BLACK: Kc7 (BLACK to move)
  17. The KNLvK endgame has 13 fortress draws, i.e. positions where the King can keep the pieces occupied forever. They are all in a single connected set:
  18. The number of fortress draws in the KNWvK endgame is very unusual (4.29%), being neither very large nor very small. This means there are plenty of pitfalls for WHITE in this more-or-less won endgame.
  19. The KLWvK endgame has 567 fortress draws, i.e. positions where the King can keep the pieces occupied forever. The vast majority of these positions have the Wazir in the corner with the Black King between it and the White King. The Camel is stuck protecting the Wazir while being harassed by the Black King. The remaining positions fall into two sets:
  20. Two Wazirs cannot checkmate a lone King.
  21. Two Ferz's, even on different colors, cannot checkmate a lone King.
  22. A Wazir and a Ferz cannot checkmate a lone King.
  23. Two Camels, even on different colors, cannot checkmate a lone King.
  24. A Camel and a Zebra cannot checkmate a lone King.
  25. A Camel and a Wazir can checkmate a lone King, which is surprising. It can take as long as 77 moves (154 halfmoves) to force mate. I've played through this endgame a few times, and it appears that the corner colored opposite the Camel is "safe" for the lone King: Even when the White King and both pieces are bearing down on the lone King in this corner, is still takes over 40 moves (80 halfmoves) to force him into the "deadly" corner and checkmate him. Checkmate seems always to be given by the Wazir on the square next to the deadly corner.
  26. A Knight and a Camel can checkmate a lone King.
  27. A Knight and a Zebra cannot checkmate a lone King.
  28. A Knight and a Wazir can checkmate a lone King.
The 3-piece endgames have been delayed due to a faulty hard disk. I need to do a few more standard chess endgames (KRNvKR, KQNvKQ, KQBvKQ), and then I'll start on endgames with fantasy pieces. (One final note: The endgames KNvKW, KLvKW, and KZvKW all show an unusually large number of perpetual-check draws. This is a side affect of the way my program classifies draws: The strong side never wants to allow simplification, even if he's the one doing the capturing. As a result, the weak side can check the strong side's King without fear of being captured. I intend to fix this "anomaly" soon.)

Sincerely, 

Dave McCooey                   Email: dmccooey at earthlink dot net

(8x8 SQUARE BOARD)

Endgames: [King + One Piece] vs [King + One Piece]

(A)mazon, (Q)ueen, (C)hancellor, Pe(G)asus, (R)ook, (B)ishop, K(N)ight Kn(I)ghtrider, Came(L), (Z)ebra, (W)azir, (F)erz

(d) = colorbound pieces on different colored squares
(s) = colorbound pieces on same colored squares 

                   ------------------------------------------------------------
                   |||||||||||||||||   Percentage Breakdown   |||||||||||||||||
Two     Unique     ------------------------------------------------------------
Piece   Legal      WHITE     BLACK     Simple    Fortress  Perpetual Stale-
Endgame Positions  Wins      Wins      Draws     Draws     Checks    mates
------- ---------  --------- --------- --------- --------- --------- ---------
KAvKA     1957100  38.011241 38.011241  1.257166 22.720352  0.000000  0.000000
KAvKQ     2097766  47.066260 22.256105  8.358368  4.568908 17.750359  0.000000
KAvKC     2185790  57.044089 19.866044  6.802255  0.001052 16.285279  0.001281
KAvKG     2317082  78.143933 16.387335  2.876463  0.000000  2.583249  0.009020
KAvKR     2326456  79.972671 16.618539  3.353986  0.000000  0.053515  0.001290
KAvKI     2435852  83.654754  0.000000 16.333628  0.000000  0.001149  0.010469
KAvKB     2457748  86.003122  0.000000 13.987398  0.000000  0.000000  0.009480
KAvKN     2545772  89.224526  0.000000 10.764711  0.000000  0.000000  0.010763
KAvKL     2569013  89.260934  0.000000 10.728439  0.000000  0.000000  0.010627
KAvKZ     2585666  89.952801  0.000000 10.036060  0.000000  0.000000  0.011138
KAvKW     2590546  90.661814  0.000000  9.336680  0.000000  0.000000  0.001505
KAvKF     2602900  91.039072  0.000000  8.951208  0.000000  0.000000  0.009720
KQvKQ     2238432  21.099815 21.099815  0.558248 57.242123  0.000000  0.000000
KQvKC     2326456  31.175488 26.860168 12.005170 18.808050 11.151124  0.000000
KQvKG     2457748  34.440594 19.463590  4.469539 38.102360  3.523673  0.000244
KQvKR     2467122  80.708858 15.745269  3.461483  0.000000  0.084390  0.000000
KQvKI     2576518  81.378123  0.000000 18.619082  0.000000  0.002562  0.000233
KQvKB     2598414  86.704159  0.000000 13.295379  0.000000  0.000000  0.000462
KQvKN     2686438  88.341216  0.000000 11.658188  0.000000  0.000372  0.000223
KQvKL     2709679  89.720812  0.000185 10.278561  0.000000  0.000000  0.000443
KQvKZ     2726332  90.403847  0.000220  9.595493  0.000000  0.000000  0.000440
KQvKW     2731212  91.143419  0.000000  8.855812  0.000000  0.000769  0.000000
KQvKF     2743566  91.507075  0.000000  8.492488  0.000000  0.000000  0.000437
KCvKC     2414480  18.969592 18.969592  0.730841 61.329976  0.000000  0.000000
KCvKG     2545772  35.445005 21.255281  3.691847 32.256266  7.350265  0.001336
KCvKR     2555146  44.564851 15.506315  3.659086 36.202941  0.066806  0.000000
KCvKI     2664542  84.389625  0.000000 15.584404  0.000000  0.025708  0.000263
KCvKB     2686438  85.873264  0.000000 14.126140  0.000000  0.000000  0.000596
KCvKN     2774462  89.931706  0.000000 10.064618  0.000000  0.003424  0.000252
KCvKL     2797703  89.925164  0.000000 10.074336  0.000000  0.000000  0.000500
KCvKZ     2814356  90.594331  0.000000  9.405171  0.000000  0.000000  0.000497
KCvKW     2819236  91.309312  0.000000  8.690688  0.000000  0.000000  0.000000
KCvKF     2831590  91.680010  0.000000  8.319425  0.000000  0.000000  0.000565
KGvKG     2677064  15.160116 15.160116  0.539322 69.140446  0.000000  0.000000
KGvKR     2686438  25.785631 17.818688  5.470664 50.888723  0.034953  0.001340
KGvKI     2795834  84.690400  0.000000 15.267931  0.000000  0.040382  0.001288
KGvKB     2817730  87.354395  0.000106 12.645498  0.000000  0.000000  0.000000
KGvKN     2905754  90.317487  0.000000  9.675905  0.000000  0.005300  0.001308
KGvKL     2928995  90.234978  0.000205  9.763656  0.000000  0.000000  0.001161
KGvKZ     2945648  90.922303  0.000204  9.076169  0.000000  0.000000  0.001324
KGvKW     2950528  91.629566  0.000000  8.369112  0.000000  0.000000  0.001322
KGvKF     2962882  92.020911  0.000135  7.978954  0.000000  0.000000  0.000000
KRvKR     2695812  14.894102 14.894102  0.211958 69.999837  0.000000  0.000000
KRvKI     2805208  18.000412  0.000927 18.126000 63.838867  0.033580  0.000214
KRvKB     2827104  18.435226  0.000000 13.635756 67.928594  0.000000  0.000424
KRvKN     2915128  28.203358  0.000172 11.119203 60.673219  0.003842  0.000206
KRvKL     2938369  51.211165  0.000170  9.702185 39.086071  0.000000  0.000408
KRvKZ     2955022  90.960338  0.000372  9.038884  0.000000  0.000000  0.000406
KRvKW     2959902  91.713408  0.000068  8.283653  0.000777  0.002095  0.000000
KRvKF     2972256  92.064916  0.000000  7.934680  0.000000  0.000000  0.000404
KIvKI     2914604   0.000995  0.000995  0.000000 99.998010  0.000000  0.000000
KIvKB     2936500   0.000988  0.000102 14.389750 85.609161  0.000000  0.000000
KIvKN     3024524   0.000959  0.000198 11.083430 88.867108  0.048305  0.000000
KIvKL     3047765   0.002362  0.000197  9.580463 90.416978  0.000000  0.000000
KIvKZ     3064418   0.001273  0.000359  8.893858 91.104510  0.000000  0.000000
KIvKW     3069298   0.002346  0.000098  8.601869 87.273116  4.122571  0.000000
KIvKF     3081652   0.000941  0.000130  8.027188 91.971741  0.000000  0.000000
KBvKB     2958396   0.002231  0.002231  0.000068 99.995471  0.000000  0.000000
KBvKN     3046420   0.000098  0.000197 21.324998 78.674707  0.000000  0.000000
KBvKL(d)  1553412   0.004249  0.001352  5.328271 94.666128  0.000000  0.000000
KBvKL(s)  1516249   0.000000  0.000000  0.000000   (100)    0.000000  0.000000
KBvKZ     3086314   0.000486  0.000389  9.123278 90.875847  0.000000  0.000000
KBvKW     3091194   0.000000  0.000000  0.006308 99.993692  0.000000  0.000000
KBvKF(d)  1570937   0.004201  0.000446  5.324784 94.670569  0.000000  0.000000
KBvKF(s)  1532611   0.000000  0.000000  0.000130 99.999870  0.000000  0.000000
KNvKN     3134444   0.000191  0.000191  0.000000 99.999617  0.000000  0.000000
KNvKL     3157685   0.000507  0.000190 10.244087 89.755216  0.000000  0.000000
KNvKZ     3174338   0.000504  0.000347  9.877934 90.121216  0.000000  0.000000
KNvKW     3179218   0.000503  0.000094 11.152994  0.000000 88.846408  0.000000
KNvKF     3191572   0.000188  0.000125  9.388101 90.611586  0.000000  0.000000
KLvKL(d)  1613456   0.001302  0.001302  0.000000 99.997397  0.000000  0.000000
KLvKL(s)  1567470   0.000000  0.000000  0.000000   (100)    0.000000  0.000000
KLvKZ     3197579   0.000657  0.000375  8.694734 91.304234  0.000000  0.000000
KLvKW     3202459   0.000656  0.000156 10.209998  0.000000 89.789190  0.000000
KLvKF(d)  1630981   0.001288  0.000429  5.225015 94.773268  0.000000  0.000000
KLvKF(s)  1583832   0.000000  0.000000  0.000000   (100)    0.000000  0.000000
KZvKZ     3214232   0.000684  0.000684  0.000000 99.998631  0.000000  0.000000
KZvKW     3219112   0.000683  0.000155 13.695268  0.000000 86.303894  0.000000
KZvKF     3231466   0.000681  0.000217 10.409053 89.590050  0.000000  0.000000
KWvKW     3223992   0.000155  0.000155  0.012717 99.986973  0.000000  0.000000
KWvKF     3236346   0.000000  0.000000  0.006489 99.993511  0.000000  0.000000
KFvKF(d)  1648506   0.000425  0.000425  0.000000 99.999151  0.000000  0.000000
KFvKF(s)  1600194   0.000000  0.000000  0.000125 99.999875  0.000000  0.000000

Two Piece Endgame Longest WHITE win Longest BLACK win
- (halfmoves) (halfmoves)
KAvKA 21 21
KAvKQ 60 13
KAvKC 46 13
KAvKG 44 9
KAvKR 24 5
KAvKI 12 0
KAvKB 10 0
KAvKN 10 0
KAvKL 8 0
KAvKZ 8 0
KAvKW 16 0
KAvKF 8 0
KQvKQ 19 19
KQvKC 24 23
KQvKG 41 9
KQvKR 62 5
KQvKI 52 0
KQvKB 24 0
KQvKN 38 0
KQvKL 20 1
KQvKZ 18 1
KQvKW 26 0
KQvKF 20 0
KCvKC 20 20
KCvKG 32 16
KCvKR 79 5
KCvKI 33 0
KCvKB 26 0
KCvKN 28 0
KCvKL 16 0
KCvKZ 14 0
KCvKW 20 0
KCvKF 22 0
KGvKG 17 17
KGvKR 27 11
KGvKI 113 0
KGvKB 64 1
KGvKN 43 0
KGvKL 24 1
KGvKZ 20 1
KGvKW 26 0
KGvKF 26 1
KRvKR 7 7
KRvKI 17 1
KRvKB 36 0
KRvKN 54 1
KRvKL 70 1
KRvKZ 50 1
KRvKW 46 1
KRvKF 40 0
KIvKI 1 1
KIvKB 1 1
KIvKN 1 1
KIvKL 1 1
KIvKZ 1 1
KIvKW 1 1
KIvKF1 1
KBvKB1 1
KBvKN1 1
KBvKL(d) 1 1
KBvKL(s) 0 0
KBvKZ1 1
KBvKW0 0
KBvKF(d) 1 1
KBvKF(s) 0 0
KNvKN1 1
KNvKL1 1
KNvKZ1 1
KNvKW1 1
KNvKF1 1
KLvKL(d) 1 1
KLvKL(s) 0 0
KLvKZ1 1
KLvKW1 1
KLvKF(d) 1 1
KLvKF(s) 0 0
KZvKZ 1 1
KZvKW 1 1
KZvKF 1 1
KWvKW 1 1
KWvKF 0 0
KFvKF(d) 1 1
KFvKF(s) 0 0

(8x8 SQUARE BOARD)

Endgames: [King + Two Pieces] vs [King]

(A)mazon, (Q)ueen, (C)hancellor, Pe(G)asus, (R)ook, (B)ishop, K(N)ight Kn(I)ghtrider, Came(L), (Z)ebra, (W)azir, (F)erz

(d) = colorbound pieces on different colored squares
(s) = colorbound pieces on same colored squares 


                   ------------------------------------------------------------
                   |||||||||||||||||   Percentage Breakdown   |||||||||||||||||
Two     Unique     ------------------------------------------------------------
Piece   Legal      WHITE     BLACK     Simple    Fortress  Perpetual Stale-
Endgame Positions  Wins      Wins      Draws     Draws     Checks    mates
------- ---------  --------- --------- --------- --------- --------- ---------
KAAvK     1125175  98.943809  0.000000  0.000000  0.000000  0.000000  1.056191
KAQvK     2326649  98.758730  0.000000  0.000000  0.000000  0.000000  1.241270
KACvK     2385238  98.799826  0.000000  0.000000  0.000000  0.000000  1.200174
KAGvK     2473365  99.373930  0.000000  0.000000  0.000000  0.000000  0.626070
KARvK     2462480  99.456889  0.000000  0.000000  0.000000  0.000000  0.543111
KAIvK     2547214  93.904124  0.000000  5.593170  0.000000  0.000000  0.502706
KABvK     2550607  93.785401  0.000000  5.715894  0.000000  0.000000  0.498705
KANvK     2609196  93.781763  0.000000  5.894919  0.000000  0.000000  0.323318
KALvK     2623313  93.692556  0.000000  5.968788  0.000000  0.000000  0.338656
KAZvK     2632666  93.673789  0.000000  6.020399  0.000000  0.000000  0.305812
KAWvK     2627928  93.600510  0.000000  6.110517  0.000000  0.000000  0.288973
KAFvK     2637776  93.597068  0.000000  6.112612  0.000000  0.000000  0.290320
KQQvK     1208031  98.537124  0.000000  0.000000  0.000000  0.000000  1.462876
KQCvK     2483526  98.677002  0.000000  0.000000  0.000000  0.000000  1.322998
KQGvK     2581813  99.226861  0.000000  0.000000  0.000000  0.000000  0.773139
KQRvK     2571940  99.312426  0.000000  0.000000  0.000000  0.000000  0.687574
KQIvK     2669137  94.102176  0.000000  5.337680  0.000000  0.000000  0.560144
KQBvK     2670227  94.002495  0.000000  5.459835  0.000000  0.000000  0.537670
KQNvK     2738690  93.987417  0.000000  5.616189  0.000000  0.000000  0.396394
KQLvK     2753099  93.924192  0.000000  5.687409  0.000000  0.000000  0.388399
KQZvK     2763786  93.922503  0.000000  5.734778  0.000000  0.000000  0.342718
KQWvK     2760115  93.846017  0.000000  5.817874  0.000000  0.000000  0.336109
KQFvK     2770649  93.855266  0.000000  5.819467  0.000000  0.000000  0.325267
KCCvK     1271038  98.861875  0.000000  0.000000  0.000000  0.000000  1.138125
KCGvK     2660975  99.335056  0.000000  0.000000  0.000000  0.000000  0.664944
KCRvK     2639391  99.260966  0.000000  0.000000  0.000000  0.000000  0.739034
KCIvK     2739315  94.237026  0.000000  5.200935  0.000000  0.000000  0.562038
KCBvK     2759263  94.212259  0.000000  5.283657  0.000000  0.000000  0.504084
KCNvK     2816840  94.172441  0.000000  5.460374  0.000000  0.000000  0.367185
KCLvK     2834623  94.059598  0.000000  5.523839  0.000000  0.000000  0.416563
KCZvK     2846260  94.110587  0.000000  5.568606  0.000000  0.000000  0.320807
KCWvK     2842870  94.045419  0.000000  5.648517  0.000000  0.000000  0.306064
KCFvK     2857898  94.040585  0.000000  5.641804  0.000000  0.000000  0.317611
KGGvK     1375726  99.650875  0.000000  0.000000  0.000000  0.000000  0.349125
KGRvK     2769423  99.745109  0.000000  0.000000  0.000000  0.000000  0.254891
KGIvK     2851166  94.797357  0.000000  4.997008  0.000000  0.000000  0.205635
KGBvK     2858810  94.703775  0.000000  5.099674  0.000000  0.000000  0.196550
KGNvK     2937972  94.642597  0.000000  5.235346  0.000000  0.000000  0.122057
KGLvK     2957255  94.590389  0.000000  5.294876  0.000000  0.000000  0.114735
KGZvK     2970290  94.567298  0.000000  5.336179  0.000000  0.000000  0.096523
KGWvK     2972755  94.508226  0.000000  5.401723  0.000000  0.000000  0.090051
KGFvK     2979243  94.486485  0.000000  5.412012  0.000000  0.000000  0.101502
KRRvK     1374940  99.820501  0.000000  0.000000  0.000000  0.000000  0.179499
KRIvK     2861238  94.834054  0.000000  4.979313  0.000000  0.000000  0.186632
KRBvK     2878883  94.781379  0.000000  5.064117  0.000000  0.000000  0.154504
KRNvK     2946334  94.678506  0.000000  5.220386  0.000000  0.000000  0.101109
KRLvK     2964409  94.606547  0.000000  5.281997  0.000000  0.000000  0.111456
KRZvK     2977380  94.610765  0.000000  5.323372  0.000000  0.000000  0.065863
KRWvK     2975057  94.539332  0.000000  5.397544  0.000000  0.000000  0.063125
KRFvK     2990771  94.550937  0.000000  5.391152  0.000000  0.000000  0.057911
KIIvK     1473381  90.211765  0.000000  9.709437  0.000000  0.000000  0.078798
KIBvK     2973089  90.063500  0.000000  9.862268  0.000000  0.000000  0.074233
KINvK     3043267  90.100704  0.000000  9.845833  0.000000  0.000000  0.053462
KILvK     3066558  89.939665  0.000000 10.011126  0.000000  0.000000  0.049208
KIZvK     3080730  89.877529  0.000000 10.075567  0.000000  0.000000  0.046904
KIWvK     3084188  89.749522  0.000000 10.204598  0.000000  0.000000  0.045879
KIFvK     3095018  89.592468  0.000000 10.348631  0.016995  0.000000  0.041906
KBBvK      746684  89.021058  0.000000 10.889881  0.000000  0.000000  0.089060
KBNvK     3067466  89.704466  0.000000 10.243015  0.000000  0.000000  0.052519
KBLvK(d)  1553412  88.722245  0.000000 11.219174  0.000966  0.000000  0.057615
KBLvK(s)  1533629   0.000000  0.000000  8.495797 91.453018  0.000000  0.051186
KBZvK     3101410  90.084736  0.000000  9.867222  0.000000  0.000000  0.048043
KBWvK     3104942  89.593461  0.000000 10.353076  0.000064  0.000000  0.053399
KBFvK(d)  1570937  87.340549  0.000000 12.608335  0.000000  0.000000  0.051116
KBFvK(s)  1541179   0.000000  0.000000  8.642409 91.309900  0.000000  0.047691
KNNvK     1573368   0.005847  0.000000 10.502375 89.461016  0.000000  0.030762
KNLvK     3168565  89.491110  0.000000 10.475689  0.000410  0.000000  0.032791
KNZvK     3183884   0.006376  0.000000 10.660941 89.307022  0.000000  0.025660
KNWvK     3187697  79.122796  0.000000 16.561329  4.290558  0.000000  0.025316
KNFvK     3199365   2.363594  0.000000 17.885205 79.725446  0.000000  0.025755
KLLvK(d)   806728   0.019585  0.000000 11.280258 88.665696  0.000000  0.034460
KLLvK(s)   787969   0.000000  0.000000  8.858217 91.104980  0.000000  0.036803
KLZvK     3205579   0.007362  0.000000  9.982627 89.979595  0.000000  0.030416
KLWvK     3209471  88.411081  0.000000 11.540999  0.017666  0.000000  0.030254
KLFvK(d)  1630981   0.010301  0.000000 14.310406 85.648760  0.000000  0.030534
KLFvK(s)  1590215   0.000000  0.000000 10.455567 89.513745  0.000000  0.030688
KZZvK     1610472   0.004843  0.000000 10.341689 89.630990  0.000000  0.022478
KZWvK     3225089   0.009116  0.000000 11.394135 88.575013  0.000000  0.021736
KZFvK     3236871   1.494468  0.000000 11.828522 86.652140  0.000000  0.024870
KWWvK     1614603   0.010653  0.000000 33.685061 66.282609  0.000000  0.021677
KWFvK     3241410   0.924659  0.000000 36.766407 62.283790  0.000000  0.025143
KFFvK(d)   824253   0.004610  0.000000 30.115632 69.853310  0.000000  0.026448
KFFvK(s)   802344   0.000000  0.000000 26.404136 73.568694  0.000000  0.027170

Two
Piece    Longest WHITE    Longest BLACK
Endgame  win (halfmoves)  win (halfmoves)
-------  ---------------  ---------------
KAAvK           5                0
KAQvK           5                0
KACvK           5                0
KAGvK           5                0
KARvK           5                0
KAIvK           6                0
KABvK           8                0
KANvK           8                0
KALvK           8                0
KAZvK           8                0
KAWvK           8                0
KAFvK          10                0
KQQvK           7                0
KQCvK           7                0
KQGvK           9                0
KQRvK           9                0
KQIvK          13                0
KQBvK          13                0
KQNvK          15                0
KQLvK          16                0
KQZvK          17                0
KQWvK          20                0
KQFvK          18                0
KCCvK           7                0
KCGvK           9                0
KCRvK           9                0
KCIvK          14                0
KCBvK          15                0
KCNvK          16                0
KCLvK          20                0
KCZvK          20                0
KCWvK          24                0
KCFvK          22                0
KGGvK          11                0
KGRvK          13                0
KGIvK          19                0
KGBvK          21                0
KGNvK          23                0
KGLvK          25                0
KGZvK          27                0
KGWvK          34                0
KGFvK          30                0
KRRvK          11                0
KRIvK          22                0
KRBvK          24                0
KRNvK          24                0
KRLvK          26                0
KRZvK          28                0
KRWvK          29                0
KRFvK          30                0
KIIvK          44                0
KIBvK          44                0
KINvK          54                0
KILvK          56                0
KIZvK          61                0
KIWvK          60                0
KIFvK          84                0
KBBvK          38                0
KBNvK          66                0
KBLvK(d)       66                0
KBLvK(s)        0                0
KBZvK          78                0
KBWvK          86                0
KBFvK(d)       74                0
KBFvK(s)        0                0
KNNvK           1                0
KNLvK          98                0
KNZvK           1                0
KNWvK          90                0
KNFvK          72                0
KLLvK(d)        3                0
KLLvK(s)        0                0
KLZvK           3                0
KLWvK         154                0
KLFvK(d)        3                0
KLFvK(s)        0                0
KZZvK           1                0
KZWvK           1                0
KZFvK          93                0
KWWvK           1                0
KWFvK          91                0
KFFvK(d)        3                0
KFFvK(s)        0                0
The one-piece endgame results are as one can predict: The Amazon, Chancellor, and Pegasus can force mate, while the other 5 cannot. In fact, the other 5 fantasy pieces can't checkmate even if the lone King cooperates! (None of these pieces can cover two adjacent squares.)

The maximum number of moves that it can take the Amazon, Chancellor, or Pegasus to checkmate are as follows:

Amazon: 4
Chancellor: 11
Pegasus: 17

For comparison:
Queen: 10
Rook: 16

The database file created by my program for a given 2-piece endgame contains all simplified versions of the endgame as well. So, it is possible to play a King + Pegaus vs King endgame using any of the 2-piece endgame databases that contain the Pegasus piece. What I don't have, however, is a percentage breakdown for 1-piece endgames.

See also:

Written by: Dave McCooey.
WWW page created: July 16, 1996. Last modified: September 5, 2001.