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H. G. Muller wrote on Tue, Jul 8, 2008 07:15 PM UTC:Betza's HFD (= (1,1)+(2,0)+(3,0) leaper) can mate a bare King (with the aid of its own King) even on a 14x14 board. With white to move the K + HFD vs K end-game is 100% won. There really isn't a single position that is not won. (Some KXK end-games have a few draws when the X is trapped in a corner by the bare King, or when they cannot lift a stalemate condition of a cornered King that attacks the X. But not when X = Half Duck) The number of moves it takes is: 14x14: 94 12x12: 66 10x10: 42 8x8: 27 On 16x16 it is usually draw: only 7.78% of the positions with white to move is won. The longest win on 16x16 still takes 65 moves, though, from the following position: k . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . 15 . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . 13 . . . . . K . . . . . . . . . . 12 . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . 3 . . H . . . . . . . . . . . . . 2 . . . . . . . . . . . . . . . . 1 a b c d e f g h i j k l m n o p White to move, mate in 65. Solution: 1. Hc5, Kb15 2. Hc8, Kc14 3. Hc11+, Kd14 4. Kf13, Ke15 5. Kg14, Ke14 6. He11+, Ke15 7. Kg15, Kd14 8. Kf14, Kd13 9. Kf13, Kc13 10. Ke13, Kc12 11. Ke12, Kc13 12. Hd10, Kd14 13. Hd13, Ke15 14. Kf13, Kf15 15. Hg13, Ke15 16. Hh12, Kf15 17. Hh14, Kf16 18. Kf14, Kg16 19. Kg14, Kf16 20. He14, Kg16 21. He16+, Kf16 22. Hf15, Ke15 23. Hf13, Kd14 24. Kf14, Kc14 25. Ke13, Kb13 26. He12, Kc13 27. Hd11, Kc14 28. Hd14, Kb13 29. Kd12, Kb12 30. Hd11, Kb13 31. He10, Kb12 32. Hc10, Ka12 33. Hb9, Kb13 34. Kd13, Ka12 35. Kc12, Ka11 36. Kc11, Ka12 37. Hb12, Kb13 38. Hd12, Kc14 39. Kc12, Kc15 40. Kd13, Kd16 41. He13, Kc15 42. Hf14, Kd15 43. Hg15+, Kc15 44. Kc13, Kd16 45. Kd14, Ke16 46. Ke14, Kd16 47. Hd15, Kc15 48. Hd13, Kb14 49. Kd14, Kb15 50. Hb13+, Kb14 51. Hb11+, Kb15 52. Kd15, Ka14 53. Kc14, Ka13 54. Kc13, Ka14 55. Hb14, Kb15 56. Hd14, Kb16 57. Kc14, Ka14 58. He15, Kb16 59. Hb15, Ka15 60. Hb13, Ka16 61. Kc15, Ka15 62. Ha12+, Ka16 63. Ha14+, Ka15 64. Ha11, Ka16 65. Ha13#
H. G. Muller wrote on Wed, Jul 9, 2008 10:27 AM UTC:Some more data about mating potential of the short-range leapers:
Betza NR NAME Longest mate (if generally won)
8x8 10x10 12x12 14x14 16x16
F 4 Ferz color bound
W 4 Wazir pure alternator
A 4 Alfil color bound
D 4 Dabbaba color bound
N 8 Knight pure alternator
FW 8 Commoner 18 29 49 62 -
FA 8 modern Elephant color bound
FD 8 ? color bound
WA 8 Waffle no mates
WD 8 Woody Rook 29 52 -
AD 8 Alibaba color bound
FN 12 ? 22 32 44 59 100
WN 12 Vicar pure alternator
AN 12 Kangaroo 35 63 -
DN 12 Carpenter 31 44 62 92 -
FWA 12 Crowned Alfil 15 22 31 41 53
FWD 12 Crowned Dabbaba 15 20 27 33 40
FAD 12 ? color bound
WAD 12 ? 26 39 -
FWN 16 Centaur 13 17 21 28 33
FAN 16 High Priestess 17 23 30 36 45
FDN 16 ? 14 19 25 31 38
WAN 16 ? 22 31 43 57 74
WDN 16 Minister 17 23 30 36 45
ADN 16 Squirrel 19 24 31 38 46
FWAD 16 Mastodon 13 19 24 29 36
FWADN 24 Lion 5 7 9 10 12
Note that the Lion does not need King assistence to perform the checkmate,
which is why it can be so fast. It is easy to prove it can mate on boards
of any size, and indeed on an infinitely large board (which is not the
same!). It does not even need a corner, just an edge.
'no mates' means that the piece does not cover two orthogonally
neighboring squares, which is the minimum requirement to create a
checkmate position (in a corner). Being color-bound, or a pure color
alternator implies this.Some fascinating results in this thread! NOTE: Betza never gave the FAD a proper name, in Chess With Different Armies, but Daniel C. Macdonald calls the WAD a Champion in Omega Chess.
The Buffalo and the Gryphon or Griffon are used in Gigachess, a 14x14 variant by Jean-Louis Cazaux. They should each be able to checkmate, but I have never seen a computer proof. The Gryphon, FA, WD, and FWADN also appear in his new 12x12 variant: Balance 12.
EDIT: Nice to see exact numbers for the Buffalo mates. I realised that both Bison and Buffalo can checkmate just after I posted. And I was thinking of Eric Greenwood's Cavalier piece when I asked about Gryphon mates. The Cavalier is a sort of multipath Gryphon, which cannot stop on any adjacent square. Benjamin C. Good wrote (March 13, 2002) that this piece cannot, in general, force mate - because it does not attack adjacent squares.
H. G. Muller wrote on Tue, Jul 15, 2008 02:04 PM UTC:The Buffalo is upward compatible with the Bison, and adds the Knight moves
to it. Although this does not endow it with more speed, it helps
tremendously in accelerating the checkmating of a bare King. The long
stride of the Bison makes for very awkward manouevring. The Bison mates
are very tedious, the average mate is only some 10 moves shorter than the
longest mate. On 14x14, of the 18.5M positions (with the white King in a
given quadrant), only some 100,000 have a DTM < 60. After that it
explodes, the most common DTM shared by 203,408 positions being 73.
Apparently there is a very easy initial phase, probably just walking the
winning King to the center, driving away the bare King from there with the
aid of the Bison. But after that, a very painstaking drive towards the
corner starts, in which the Bison can only barely prevent that the bare
King nescapes back into the open.
The extra Knight move of the Buffalo allows you to cutt off the bare King
much more efficiently:
8x8 10x10 12x12 14x14 16x16
Bison: (CZ) 27 40 55 82 -
Buffalo: (CNZ) 18 24 31 38 45
The remaing Camel/Knight/Zebra compound, the GNU or Wildebeest (NZ), has
no mating potential. There are only 2 irreducible checkmate positions, and
they cannot be enforced on any size board. Similar to KNNK, the bare King
would voluntarily have to step into a mated-in-1 position.
For the Griffon no computer is needed to give the proof. The system is
similar to the Rook, and works even in the corner of an infinite board.
(So certainly for boards of any size.) It is even easier, because the
Griffon immediately traps the bare King in a corner, without the latter
being able to attack it, like it could do for a Rook.
In fact, with the Griffon there is even a much faster method than with the Rook, as a Griffon can trap the bare King in a narrow corridor, its own King acting as a piston to push the bare King to the edge.H. G. Muller wrote on Thu, Jul 17, 2008 09:30 PM UTC:'The Cavalier is a sort of multipath Gryphon, which cannot stop on any adjacent square. Benjamin C. Good wrote (March 13, 2002) that this piece cannot, in general, force mate - because it does not attack adjacent squares.' Well, pieces that do not attack at least two orthogonally adjacent squares obviously can not checkmate. But the Cavalier obiously attacks lots of adjacent squares. I think the percieed problem was that the rays covered b the Cavalier are not 'air tight', but have a hole in them, allowing the bre King to escape its confines by approaching the Cavalier. A Cavalier, unlike a Rook, can not change to another position alog the ray it covers. The mate is very easy, though, (even on infinite boards) as a Cavalier can also do things a Rook cannot do. You don't even need a corner, just an edge (say 1st rank), and it works on an infinite board.It works like this: 1) Cut off the bare King from moving away from the edge, (a rank, say), and walk your own King to be further from the edge than he is. 2) Cut off the bare King moving laterally away from the file your own King is in, and step towards his file, staying further from the edge than he is. 3) When the Kings are nearly in the same file, position the Cavalier in the file of the bare King, so that he gets trapped in the 'corridor' between the Cavalier's attack lines. 4) Use your King to push the bare King towards the edge, walking on the same file, until he reaches 1st rank (on f1, say). 5) Lift the stalemate danger by moving your Cavalier to a file far away, so you can safely take opposition on 3rd rank. 6) We now have to get the bare King into opposition twice, once for driving him back to 1st rank with check along the rank, second time for checkmating. In both cases we shephard him into opposition by first taking opposition ourself, and when he steps sideways, cut off the file two files away from our King. He then either has to step back into opposition, or step back immediately. The main problem is keeping enough distance, as the Cavalier has thes 'holes' in its attack set nearby. So in general, when advancing towards the edge, for sideway checking, we move one file away from the bare King. On a small board this might not be possible, and we have to manouevre a bit. This takes some extra moves, but the principle remains the same. e.g.: w: Kd4, Cd8 b: Kd2 1. ... Kd1, 2. Cg7 (out of the way), Kc2 3. Ca8 (cut off b-file), Kd2 Now we would have liked to check from the side, but our Cavalier is on on the a-file, and the b-file is too close to cover c2. So we nudge him to the other side: 4. Cb6 (cut off c-file), Ke2 5. Cg7 (cut off f-file), Kd2 6. Ch3+ (got him!), Ke1 7. Ke3 (opposition), Kf1 Now we would have liked to cut off the g-file, but our Cavalier is already on the h-file, and too close to cover f2. And even if he was, we are in zugzwang. So again some delay displacing the position sideways to gain room, and then nudge him to the long side: 8. Kf3 (opposition), Ke1 (only move, g1 was attacked) 9. Cc5 (cut off d-file) , Kf1 10. Cb2#. Easy as pie...
H. G. Muller wrote on Sat, Mar 10, 2012 12:14 PM UTC:I wrote a new tablebase generator, dedicated to determine mating potential,
which does not assume any symmetry, and hence can also do odd board sizes.
Not using symmetry makes it more memory hungry, but on the other hand, it
does not need the board size to be rounded up to a power of 2, like the old
one, which can save memory. As a result I can now do 4-men end-games
(mating with a pair of pieces that do not have mating potential by
themselves) on boards upto 12x12. (This takes 421MB RAM. 4-men on 14x14
would take 1.6GB, which is more than I have.)
I updated the table I had with some selected odd board sizes, mainly to
establish exactly at which board size a leaper loses mating potential. New in
the list is the (asymmetric) Charging Knight of CwDA's Nutters army.
I also added a limited version of the Cavalier, as a Knight-Camel-Giraffe
compound. This already is enough to give it mating potential on quite
large boards. In fact, the longest mate seems to increase with board size
strictly linearly, suggesting it could do it on boards of any size.
(I tried upto 25x25, which takes 84 moves.)
The largest board on which the WAN has mating potential is 20x20 (160 moves).
For KA this is 23x23 and 208 moves. KD still has it at 25x25, but can
be proven to even have it on a quarter infinite board, which is quite
exceptional for a piece with only 12 moves. ADN also has no problems yet
at 25x25 (the biggest I could go), and take 84 moves there.
Betza NR NAME Longest mate (if generally won)
8x8 10x10 11x11 12x12 14x14 15x15 16x16 17x17
F 4 Ferz color bound
W 4 Wazir pure alternator
A 4 Alfil color bound
D 4 Dabbaba color bound
N 8 Knight pure alternator
FW 8 Commoner 18 29 35 49 62 -
FA 8 modern Elephant color bound
FD 8 ? color bound
WA 8 Waffle no mates
WD 8 Woody Rook 29 52 -
AD 8 Alibaba color bound
fNbsK 9 Charging Knight 33 52 65 82 155 -
FN 12 ? 22 32 38 44 59 72 100 -
WN 12 Vicar pure alternator
AN 12 Kangaroo 35 63 78 -
DN 12 Carpenter 31 44 52 62 92 -
FWA 12 Crowned Alfil 15 22 31 41 53
FWD 12 Crowned Dabbaba 15 20 27 33 40
FAD 12 ? color bound
WAD 12 ? 26 39 -
FWN 16 Centaur 13 17 21 28 33
FAN 16 High Priestess 17 23 30 36 45
FDN 16 ? 14 19 25 31 38
WAN 16 ? 22 31 43 57 74
WDN 16 Minister 17 23 30 36 45
ADN 16 Squirrel 19 24 31 38 46
FWAD 16 Mastodon 13 19 24 29 36
FWADN 24 Lion 5 7 9 10 12
HFD 16 Half Duck 51 66 94 107 -
CZ 16 Bison 27 40 48 55 82 104 -
CNZ 24 Buffalo 18 24 31 38 45
CNG 24 Pseudo-Cavalier 18 25 28 32 39 43 46 50
4-men: B+N 33 48 64
KBN.K is the only 4-men I tried so far on larger boards. Peculiarity here
is that B is color-bound, and mate is only possible in the corner of the B
color. This makes that it is never generally won on odd-size boards,
because all corners have the same color there. To win when B is on the
right color is easier, though, because any corner will do, so the bare King
will always be comparatively close to a deadly one. While on even-sized
boards he can take shelter in the safe corner.7 comments displayed
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[two Knights] the longest forced mate is: 1 move
[two Cannons] the longest forced mate is: 2 moves (I think)
In this thread we can discuss forced checkmates by a White King plus [one or two pieces] against a lone Black King on a 12x12 board, with stalemate counting as a draw. Three important questions: (1) How many moves does it take to mate? (2) How often can White reach an endgame position where such a mate is possible? (3) If the first two questions have favorable answers, can we hope that forced mate is usually possible on 16x16 boards, or is there a counterexample? The examples given above involve unusual endgame positions, which would normally result from a ridiculous blunder by Black. Dave McCooey has made a selection of pieces and posted elaborate tables with all possible results for [one or two pieces].
Recently H. G. Muller has posted comments on forced mates using the [Bison] and the [Carpenter]. While there is no PIECECLOPEDIA page, the Lion or Half-Duck can be found in several variants, including my Unicorn Great Chess. I have always hoped that this piece counts as 'suficient mating material' on a 12x12 board. I definitely find it easier to play with than Bisons (Camel + Zebra) or Carpenters (Knight + Dabbabah).