46:
392:
as the unique number n such that G+n is a second player win in misère play. Even if it looks very similar to the usual Grundy-value in normal play, it is not as powerful. In particular, it is not possible to deduce the misère Grundy value of a sum of games only from their respective misère Grundy
501:
In 2009, Martin
Schneider computed the misère grundy values up to the 3 × 9, 4 × 6, and 5 × 5 board. In 2010, Julien Lemoine and Simon Viennot extended these results up to the 3 × 15, 4 × 9 and 5 × 7 boards, along with the value of the
137:
play. This means that for any move by Player 1, Player 2 has a corresponding symmetric move across the horizontal and vertical axes. In a sense, player 2 'mimics' the moves made by Player 1. If Player 2 follows this strategy, Player 2 will always make the last move, and thus win the game.
345:. It allowed them to compute the Grundy values up to the 3 Ă— 20, 4 Ă— 9, 5 Ă— 9, 6 Ă— 7 and 7 Ă— 7 boards. Piotr Beling extended these results up to the 6 Ă— 9, 7 Ă— 8, and 7 Ă— 9 boards.
141:
In the even-by-odd case, the first player wins by similar symmetry play. Player 1 places their first domino in the center two squares on the grid. Player 2 then makes their move, but Player 1 can play symmetrically thereafter, thus ensuring a win for Player 1.
340:
In 2009, Martin
Schneider computed the Grundy values up to the 3 Ă— 9, 4 Ă— 5 and 5 Ă— 7 boards. In 2010, Julien Lemoine and Simon Viennot applied to the game of Cram algorithms that were initially developed for the game of
109:
which they place on the grid in turn. A player can place a domino either horizontally or vertically. Contrary to the related game of
Domineering, the possible moves are the same for the two players, and Cram is then an
72:
and the only difference in the rules is that players may place their dominoes in either orientation, but it results in a very different game. It has been called by many names, including "plugg" by
201:
The symmetry strategy implies that even-by-even boards have a Grundy value of 0, but in the case of even-by-odd boards it only implies a Grundy value greater or equal to 1.
117:
As for all impartial games, there are two possible conventions for victory: in the normal game, the first player who cannot move loses, and on the contrary, in the
94:
The game is played on a sheet of graph paper, with any set of designs traced out. It is most commonly played on rectangular board like a 6Ă—6 square or a
535:
183:
701:
509:
boards, from n=1 to n=15 is: 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1. This sequence is conjectured to be periodic of period 3.
583:
691:
352:
boards, from n=1 to n=20 is: 1, 1, 0, 1, 1, 4, 1, 3, 1, 2, 0, 1, 2, 3, 1, 4, 0, 1, 0, 2. It doesn't appear to show any pattern.
706:
133:
for normal Cram is simple for even-by-even boards and even-by-odd boards. In the even-by-even case, the second player wins by
696:
171:
355:
The table below details the known results for boards with both dimensions greater than 3. Since the value of an
603:
555:(1974). "Mathematical Games: Cram, crosscram and quadraphage: new games having elusive winning strategies".
671:
389:
512:
The adjacent table details the known misère results for boards with both dimensions greater than 3.
82:
73:
666:
617:
526:
385:
57:
564:
522:
641:
568:
552:
530:
342:
167:
111:
77:
65:
31:
685:
35:
205:
95:
45:
69:
61:
39:
653:
17:
146:
118:
174:
indicates that in the normal version any Cram position is equivalent to a
134:
130:
149:
version, because in that case it would only ensure the player that they
505:
The sequence of currently known misère Grundy values for 3 ×
99:
106:
49:
Example of a Cram game. In the normal version, the blue player wins.
622:
616:
Julien, Lemoine; Simon, Viennot (2010). "Nimbers are inevitable".
44:
395:
203:
348:
The sequence of currently known Grundy values for 3 Ă—
175:
98:, but it can also be played on an entirely irregular
384:The misère Grundy-value of a game G is defined by
371:board, we give only the upper part of the table.
76:, and "dots-and-pairs". Cram was popularized by
121:version, the first player who cannot move wins.
584:"Solving Cram Using Combinational Game Theory"
642:Computation records of normal and misère Cram
637:
635:
633:
8:
644:, Julien Lemoine and Simon Viennot web site
599:
597:
145:Symmetry play is a useless strategy in the
654:Rust software for solving impartial games
621:
536:Winning Ways for Your Mathematical Plays
184:Winning Ways for your Mathematical Plays
606:, Martin Schneider, Master thesis, 2009
544:
397:Misère Grundy values for large boards
7:
363:board is the same as the value of a
569:10.1038/scientificamerican0374-102
186:, in particular the 2 Ă—
25:
178:of a given size, also called the
105:Two players have a collection of
64:(or any type of grid). It is the
582:Uiterwijk, Jos (December 2020).
182:. Some values can be found in
1:
190:board, whose value is 0 if
723:
30:This article is about the
29:
702:Combinatorial game theory
38:version of the game, see
102:or a cylindrical board.
692:Abstract strategy games
502:6 Ă— 6 board.
707:Paper-and-pencil games
172:Sprague–Grundy theorem
50:
60:played on a sheet of
48:
672:On Numbers and Games
390:On Numbers and Games
557:Scientific American
523:Berlekamp, Elwyn R.
398:
380:Misère Grundy-value
209:
83:Scientific American
74:Geoffrey Mott-Smith
697:Mathematical games
675:. A K Peters, Ltd.
539:. A K Peters, Ltd.
396:
204:
51:
604:Das Spiel Juvavum
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498:
333:
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208:for large boards
194:is even and 1 if
166:Since Cram is an
58:mathematical game
34:of Cram. For the
27:Mathematical game
16:(Redirected from
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667:John H., Conway
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553:Gardner, Martin
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531:Guy, Richard K.
527:Conway, John H.
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518:
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377:
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164:
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127:
92:
43:
28:
23:
22:
15:
12:
11:
5:
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563:(2): 106–108.
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375:Misère version
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168:impartial game
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157:Normal version
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112:impartial game
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78:Martin Gardner
32:impartial game
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216: Ă—
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206:Grundy values
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125:Symmetry play
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656:Piotr Beling
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588:ResearchGate
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339:
336:Known values
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180:Grundy value
179:
165:
162:Grundy value
150:
144:
140:
129:The winning
128:
116:
104:
96:checkerboard
93:
81:
53:
52:
18:Cram (games)
70:Domineering
68:version of
62:graph paper
40:Domineering
686:Categories
516:References
623:1011.5841
66:impartial
669:(2000).
533:(2003).
393:values.
198:is odd.
176:nim-heap
135:symmetry
131:strategy
107:dominoes
36:partisan
343:Sprouts
100:polygon
386:Conway
170:, the
147:misère
119:misère
618:arXiv
90:Rules
56:is a
151:lose
54:Cram
565:doi
561:230
388:in
80:in
688::
632:^
596:^
586:.
559:.
529:;
525:;
495:?
477:6
472:?
454:5
449:1
431:4
426:9
329:1
311:7
306:1
288:6
283:1
265:5
260:1
242:4
237:9
153:.
114:.
86:.
626:.
620::
590:.
571:.
567::
507:n
492:?
489:?
486:1
483:-
480:-
469:?
466:1
463:1
460:2
457:-
446:1
443:1
440:0
437:0
434:0
423:8
420:7
417:6
414:5
411:4
407:m
403:n
369:n
365:m
361:m
357:n
350:n
326:3
323:1
320:-
317:-
314:-
303:0
300:5
297:0
294:-
291:-
280:1
277:1
274:2
271:0
268:-
257:0
254:3
251:0
248:2
245:0
234:8
231:7
228:6
225:5
222:4
218:m
214:n
196:n
192:n
188:n
42:.
20:)
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