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dominoes of the previous row. Thus, by induction, each of the seven pairs of consecutive rows houses an odd number of vertical dominoes, producing an odd total number. By the same reasoning, the total number of horizontal dominoes must also be odd. As the sum of two odd numbers, the total number of dominoes—vertical and horizontal—must be even. But to cover the mutilated chessboard, 31 dominoes are needed, an odd number. Another method counts the edges of each color around the boundary of the mutilated chessboard. Their numbers must be equal in any tileable region of the chessboard, because each domino has three edges of each color, and each internal edge between dominoes pairs off boundaries of opposite colors. However, the mutilated chessboard has more edges of one color than the other.
30:
22:
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with a vertex for each available chessboard square and an edge for every pair of adjacent squares; the problem is to find a system of edges that touches each vertex exactly once. As in the coloring-based proof of the impossibility of the mutilated chessboard problem, the fact that this graph has more
190:
The puzzle is impossible to complete. A domino placed on the chessboard will always cover one white square and one black square. Therefore, any collection of dominoes placed on the board will cover equal numbers of squares of each color. But any two opposite squares have the same color: both black or
77:
meeting these conditions. One proof of its impossibility uses the fact that, with the corners removed, the chessboard has 32 squares of one color and 30 of the other, but each domino must cover equally many squares of each color. More generally, if any two squares are removed from the chessboard, the
385:
that can move only one square vertically or horizontally (not diagonally). Using similar reasoning to the mutilated chessboard problem's classic solution, this wazir's tour does not exist. For example, if the initial square is white, as each move alternates between black and white squares, the final
242:
formed by the chessboard squares. The removal of any two oppositely colored squares splits this cycle into two paths with an even number of squares each. Both of these paths are easy to partition into dominoes by following them. Gomory's theorem is specific to the removal of only one square of each
202:
starts with induction. In a given tiling of the board, if a row has an odd number of squares not covered by vertical dominoes from the previous row, then an odd number of vertical dominoes must extend into the next row. The first row trivially has an odd number of squares (namely, 7) not covered by
303:
and asking for a system that can automatically determine the unsolvability of this formulation. Most considerations of this problem provide solutions "in the conceptual sense" that do not apply to McCarthy's logic formulation of the problem. Despite the existence of general methods such as those
316:
systems that can manage the equivalences between different representations. Short proofs are possible using resolution with additional variables, or in stronger proof systems allowing the expression of avoidable tiling patterns that can prune the search space. Higher-level
390:, which asks for a tour in which the positions of certain squares match given clues. The impossibility of a corner-to-corner tour shows the impossibility of a Numbrix puzzle with the clues 1 in one corner and 64 in the opposite corner.
907:
Bilalić, Merim; Graf, Mario; Vaci, Nemanja; Danek, Amory H. (August 2019), "When the solution is on the doorstep: Better solving performance, but diminished aha! experience for chess experts on the mutilated checkerboard problem",
191:
both white. If they are removed, there will be fewer squares of that color and more of the other color, making the numbers of squares of each color unequal and the board impossible to cover. The same idea shows that no
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1111:
243:
color. Removing larger numbers of squares, with equal numbers of each color, can result in a region that has no domino tiling, but for which coloring-based impossibility proofs do not work.
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most treatments of the problem in the literature solve it in the conceptual sense, but do not actually provide proofs of the theorem in either of McCarthy's original formulations
214:
through the squares into one (left) or two (right) paths through an even number of squares, allowing the modified chessboard to be tiled by dominoes laid along the paths.
1987:
386:
square of any complete tour is black. However, the opposite corner square is white. This sort of tour of a chessboard also forms the basis of a type of puzzle called
210:
Gomory's theorem: Removing any two oppositely-colored squares of a chessboard leaves a region that can be tiled by dominoes. The two removed squares partition a
1556:
Theorem
Proving with Analytic Tableaux and Related Methods, 5th International Workshop, TABLEAUX '96, Terrasini, Palermo, Italy, May 15–17, 1996, Proceedings
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starting at a corner square of an ordinary chessboard can visit every square exactly once, and finish at the opposite corner square. The wazir is a
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rest can be tiled by dominoes if and only if the removed squares are of different colors. This problem has been used as a test case for
133:(1946), with a hint at the coloring-based solution to its impossibility. It was popularized in the 1950s through later discussions by
1905:
811:
Mathematical
Knowledge Management, 4th International Conference, MKM 2005, Bremen, Germany, July 15-17, 2005, Revised Selected Papers
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If two squares of opposite colors are removed, then the remaining board can always be tiled with dominoes; this result is
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Bivens, Irl C.; Holshouser, Arthur L.; Klein, Benjamin G. (October 2008), "Wazir circuits on an obstructed chessboard",
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can exist whenever any two squares of the same color (not just the opposite corners) are removed from the chessboard.
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A region of the chessboard that has no domino tiling, but for which coloring-based impossibility proofs do not work
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in 1937. Domino tilings also have a long history of practical use in pavement design and the arrangement of
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cuboids. The proof uses a similar chessboard-coloring argument to the mutilated chessboard problem.
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61:) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31
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are capable of handling the coloring-based impossibility proof directly; these include
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vertices of one color than the other implies that it fails the necessary conditions of
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29:
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that can automatically change to a more suitable problem representation and for
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276:, so no matching exists. The problem can also be solved by formulating it as a
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696:(February 1957), "An assortment of maddening puzzles", Mathematical Games,
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1084:"Revisiting the mutilated chessboard or the many roles of a picture"
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1234:(2021), "Conway's influence on the study of random tilings",
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255:, such as the mutilated chessboard problem, can be solved in
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based on graph matching, it is exponentially hard for
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162:The use of the mutilated chessboard problem in
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396:concerns the impossibility of packing certain
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259:, either by converting them into problems in
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360:
353:
347:
1946:from the original on 2022-07-18
1911:from the original on 2022-08-08
1729:from the original on 2022-08-08
1631:from the original on 2022-09-14
1580:from the original on 2022-07-18
1533:from the original on 2021-06-20
1474:from the original on 2022-07-18
1338:from the original on 2022-08-08
1190:The College Mathematics Journal
1167:from the original on 2018-10-23
1114:from the original on 2022-07-19
785:from the original on 2021-05-16
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278:constraint satisfaction problem
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7:
2184:Mathematical chess problems
2104:"Filling boxes with bricks"
1024:10.1007/978-3-319-06614-1_5
299:systems, formulating it in
185:
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2205:
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1250:10.1007/s00283-021-10089-3
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251:Domino tiling problems on
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1281:Watkins, John J. (2004),
1100:10.2143/LEA.255.0.3290192
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141:and Marvin Stern (1958),
116:George Stanley Rushbrooke
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1928:Rudnicki, Piotr (1996),
1617:10.1109/SFCS.2001.959896
314:knowledge representation
282:semidefinite programming
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310:artificial intelligence
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1983:
1894:10.1093/jigpal/9.3.475
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1062:10.1093/philmat/nkv008
970:10.1098/rsta.2005.1649
230:, after mathematician
223:
215:
67:
34:
26:
2072:Pi Mu Epsilon Journal
1984:
460:Erickson, Alejandro;
372:Wazir's tour problem
263:, or as instances of
221:
209:
110:dates to the work of
108:statistical mechanics
51:
32:
24:
2028:Mathematics Magazine
1967:
1356:Thurston, William P.
848:Cognitive Psychology
723:; for solution, see
448:10.1039/tf9373301272
2156:by Jay Warendorff,
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2189:Unsolvable puzzles
1979:
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157:Mathematical Games
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1216:Honsberger (1973)
1033:978-3-319-06613-4
911:Cognitive Science
843:Simon, Herbert A.
828:978-3-540-31430-1
775:(July 17, 1964),
659:978-0-333-08637-7
573:978-94-010-5542-0
501:978-3-642-14030-3
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383:fairy chess piece
370:
369:
301:first-order logic
236:Hamiltonian cycle
212:Hamiltonian cycle
172:cognitive science
135:Solomon W. Golomb
131:Critical Thinking
71:impossible puzzle
2196:
2154:Gomory's Theorem
2141:
2140:
2100:de Bruijn, N. G.
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1658:(1–3): 513–525,
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1094:(255): 289–312,
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889:(2–3): 123–138,
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422:
407:
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364:
363:
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356:
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350:
343:
337:Related problems
319:proof assistants
228:Gomory's theorem
2204:
2203:
2199:
2198:
2197:
2195:
2194:
2193:
2164:
2163:
2150:
2145:
2144:
2122:10.2307/2316785
2097:
2093:
2067:
2063:
2023:
2019:
1968:
1965:
1964:
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1957:
1949:
1947:
1926:
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1400:
1396:
1374:10.2307/2324578
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1203:10.2307/4146865
1185:
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755:. Reprinted in
726:Gardner, Martin
694:Gardner, Martin
691:
684:
670:
666:
660:
643:
639:
611:(10): 675–682,
596:
592:
586:Wayback Machine
574:
552:
545:
531:
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454:
423:
419:
414:
405:
401:
375:
374:
373:
366:
365:
358:
348:
339:
297:automated proof
280:, and applying
269:bipartite graph
257:polynomial time
249:
232:Ralph E. Gomory
200:Shmuel Winograd
188:
112:Ralph H. Fowler
96:
17:
12:
11:
5:
2202:
2192:
2191:
2186:
2181:
2176:
2174:Tiling puzzles
2162:
2161:
2149:
2148:External links
2146:
2143:
2142:
2091:
2079:(8): 505–514,
2061:
2035:(4): 276–284,
2017:
1999:(4): 573–598,
1978:
1975:
1972:
1955:
1920:
1888:(3): 475–485,
1865:
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1792:(3): 253–275,
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1505:(1–2): 37–65,
1483:
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1410:(3): 241–250,
1394:
1368:(8): 757–773,
1347:
1301:
1295:
1270:
1220:
1197:(2): 115–120,
1176:
1153:McCarthy, John
1144:
1135:Honsberger, R.
1123:
1074:
1056:(3): 295–310,
1052:, Series III,
1038:
1032:
1006:
938:
899:
873:
855:(3): 374–419,
833:
827:
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773:McCarthy, John
761:
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706:(2): 152–158,
682:
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147:Martin Gardner
95:
92:
73:: there is no
15:
9:
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4:
3:
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2179:Logic puzzles
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2014:
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959:
955:
951:
950:
942:
935:
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926:
921:
918:(8): e12771,
917:
913:
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903:
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713:
709:
705:
701:
700:
695:
689:
687:
678:
674:
673:Berge, Claude
668:
661:
655:
651:
647:
646:Gamow, George
641:
634:
630:
626:
622:
618:
614:
610:
606:
605:
600:
599:Golomb, S. W.
594:
587:
583:
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569:
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550:
548:
539:
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529:
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511:
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470:Sahni, Sartaj
467:
463:
462:Ruskey, Frank
456:
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427:
426:Fowler, R. H.
421:
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293:John McCarthy
289:
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258:
254:
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241:
237:
233:
229:
220:
213:
208:
204:
201:
196:
194:
193:domino tiling
183:
181:
177:
173:
169:
168:John McCarthy
165:
160:
158:
154:
153:
148:
144:
140:
136:
132:
128:
123:
121:
117:
113:
109:
105:
102:of grids and
101:
100:domino tiling
91:
89:
85:
81:
76:
75:domino tiling
72:
66:
64:
60:
56:
50:
48:
44:
43:tiling puzzle
40:
31:
23:
19:
2116:(1): 37–40,
2113:
2107:
2094:
2076:
2070:
2064:
2032:
2026:
2020:
1996:
1990:
1958:
1948:, retrieved
1930:
1923:
1913:, retrieved
1885:
1881:
1868:
1834:
1821:
1789:
1783:
1777:
1756:(1): 41–78,
1753:
1747:
1741:
1731:, retrieved
1703:
1697:
1681:
1655:
1649:
1643:
1633:, retrieved
1607:
1596:
1588:
1582:, retrieved
1555:
1545:
1535:, retrieved
1502:
1496:
1486:
1476:, retrieved
1465:10138/333647
1456:
1439:
1407:
1403:
1397:
1365:
1359:
1350:
1340:, retrieved
1320:
1282:
1244:(2): 40–46,
1241:
1235:
1194:
1188:
1169:, retrieved
1160:
1147:
1138:
1116:, retrieved
1091:
1087:
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1015:
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602:
593:
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537:
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327:Mizar system
290:
261:group theory
250:
227:
225:
197:
189:
161:
150:
145:(1958), and
143:Claude Berge
139:George Gamow
130:
129:in his book
124:
97:
68:
59:checkerboard
52:
38:
36:
18:
1706:(1): 3–27,
650:Puzzle-Math
579:pp. 271–274
466:Thai, My T.
253:polyominoes
104:polyominoes
2168:Categories
2086:1706.09389
1950:2022-07-18
1915:2022-07-18
1733:2022-07-19
1635:2022-07-18
1584:2022-07-18
1537:2022-07-19
1478:2022-07-18
1342:2022-07-18
1323:(1): 4–9,
1232:Propp, Jim
1171:2007-04-27
1118:2022-07-19
789:2022-07-18
534:Black, Max
412:References
306:resolution
286:relaxation
240:grid graph
159:" (1957).
122:flooring.
86:, and the
84:creativity
55:chessboard
2057:125950546
1974:×
1266:236397105
483:1103.3309
406:1 × 2 × 4
404:box with
402:6 × 6 × 6
291:In 1964,
127:Max Black
69:It is an
47:Max Black
45:posed by
2102:(1969),
2049:27643123
1944:archived
1906:archived
1861:92989148
1833:(eds.),
1724:archived
1689:(2004),
1629:archived
1625:18849777
1578:archived
1531:archived
1469:archived
1333:archived
1165:archived
1155:(1999),
1112:archived
1002:18225791
994:16188609
978:30039731
934:31446653
869:54334455
783:archived
752:24940785
720:24941903
675:(1958),
582:Archived
536:(1946),
472:(eds.),
442:: 1272,
323:Isabelle
186:Solution
155:column "
137:(1954),
63:dominoes
2138:0234841
2130:2316785
2013:1406233
1902:1828741
1814:2459540
1806:0806206
1770:0587851
1720:2070797
1674:2020358
1527:5727281
1519:1750258
1432:6753523
1424:1969128
1390:1072815
1382:2324578
1329:3323392
1258:4278473
1211:4146865
1108:4396339
1070:3404036
986:2197653
958:Bibcode
633:0067055
625:2307321
518:6603662
510:2720105
398:cuboids
388:Numbrix
238:of the
149:in his
94:History
2136:
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1900:
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325:, the
120:tatami
2126:JSTOR
2081:arXiv
2053:S2CID
2045:JSTOR
1909:(PDF)
1878:(PDF)
1857:S2CID
1810:S2CID
1727:(PDF)
1694:(PDF)
1621:S2CID
1523:S2CID
1472:(PDF)
1453:(PDF)
1428:S2CID
1378:JSTOR
1336:(PDF)
1317:(PDF)
1287:12–14
1262:S2CID
1207:JSTOR
998:S2CID
974:JSTOR
865:S2CID
748:JSTOR
716:JSTOR
621:JSTOR
514:S2CID
478:arXiv
379:wazir
331:Nqthm
284:to a
41:is a
1847:ISBN
1568:ISBN
1291:ISBN
1028:ISBN
990:PMID
930:PMID
823:ISBN
654:ISBN
568:ISBN
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114:and
57:(or
37:The
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