Jump to content

Mutilated chessboard problem: Difference between revisions

From Wikipedia, the free encyclopedia
Browse history interactively
← Previous editNext edit →
Content deleted Content added
VisualWikitext
Replace hard-to-read diagram; add another
add kerber-pollet ref, and also use it as a secondary source for McCarthy's introduction of the problem
Line 13: Line 13:
The mutilated chessboard problem itself was proposed by philosopher [[Max Black]] in his book ''Critical Thinking'' (1946), with a hint at the coloring-based solution to its impossibility.{{r|black|robinson}} It was popularized in the 1950s through later discussions by [[Solomon W. Golomb]] (1954),{{r|golomb}} [[George Gamow]] and Marvin Stern (1958),{{r|gamow-stern}} [[Claude Berge]] (1958),{{r|robinson|berge}} and [[Martin Gardner]] in his ''[[Scientific American]]'' column "[[List of Martin Gardner Mathematical Games columns|Mathematical Games]]" (1957).{{r|gardner}}
The mutilated chessboard problem itself was proposed by philosopher [[Max Black]] in his book ''Critical Thinking'' (1946), with a hint at the coloring-based solution to its impossibility.{{r|black|robinson}} It was popularized in the 1950s through later discussions by [[Solomon W. Golomb]] (1954),{{r|golomb}} [[George Gamow]] and Marvin Stern (1958),{{r|gamow-stern}} [[Claude Berge]] (1958),{{r|robinson|berge}} and [[Martin Gardner]] in his ''[[Scientific American]]'' column "[[List of Martin Gardner Mathematical Games columns|Mathematical Games]]" (1957).{{r|gardner}}


The use of the mutilated chessboard problem in [[automated reasoning]] stems from a proposal for its use by [[John McCarthy (computer scientist)|John McCarthy]] in 1964.{{r|mccarthy64}} It has also been studied in [[cognitive science]] as a test case for creative insight,{{r|kaplan-simon|akin-akin|bgvd}} Black's original motivation for the problem.{{r|black}} In the [[philosophy of mathematics]], it has been examined in studies of the nature of [[mathematical proof]].{{r|mackenzie|kerber|tanswell|starikova-vanbendegem}}
The use of the mutilated chessboard problem in [[automated reasoning]] stems from a proposal for its use by [[John McCarthy (computer scientist)|John McCarthy]] in 1964.{{r|mccarthy64|kerber-pollet}} It has also been studied in [[cognitive science]] as a test case for creative insight,{{r|kaplan-simon|akin-akin|bgvd}} Black's original motivation for the problem.{{r|black}} In the [[philosophy of mathematics]], it has been examined in studies of the nature of [[mathematical proof]].{{r|mackenzie|kerber|tanswell|starikova-vanbendegem}}


== Solution ==
== Solution ==
Line 27: Line 27:
Domino tiling problems on [[polyomino]]es, such as the mutilated chessboard problem, can be solved in [[polynomial time]], either by converting them into problems in [[group theory]]{{r|propp|thurston}} or as instances of [[bipartite matching]]. In the latter formulation, one obtains a [[bipartite graph]] 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 vertices of one color than the other implies that it fails the necessary conditions of [[Hall's marriage theorem]], so no matching exists.{{r|wright|urquhart|bpmb}} The problem can also be solved by formulating it as a [[constraint satisfaction problem]], and applying [[semidefinite programming]] to a [[Relaxation (approximation)|relaxation]].{{r|semidefinite}}
Domino tiling problems on [[polyomino]]es, such as the mutilated chessboard problem, can be solved in [[polynomial time]], either by converting them into problems in [[group theory]]{{r|propp|thurston}} or as instances of [[bipartite matching]]. In the latter formulation, one obtains a [[bipartite graph]] 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 vertices of one color than the other implies that it fails the necessary conditions of [[Hall's marriage theorem]], so no matching exists.{{r|wright|urquhart|bpmb}} The problem can also be solved by formulating it as a [[constraint satisfaction problem]], and applying [[semidefinite programming]] to a [[Relaxation (approximation)|relaxation]].{{r|semidefinite}}


In 1964, [[John McCarthy (computer scientist)|John McCarthy]] proposed the mutilated chessboard as a hard problem for [[automated proof]] systems, formulating it in [[first-order logic]] and asking for a system that can automatically determine the unsolvability of this formulation.{{r|mccarthy64}} Most considerations of this problem provide solutions "in the conceptual sense" that do not apply to McCarthy's logic formulation of the problem.{{r|andrews-bishop}} Despite the existence of general methods such as those based on graph matching, finding a solution to a logical formulation of the mutilated chessboard problem using the [[Resolution (logic)|resolution]] system of inference is exponentially hard,{{r|dantchev-riis|alekhnovich|razborov}} highlighting the need for methods in [[artificial intelligence]] that can automatically change to a more suitable problem representation.{{r|korf}} Short proofs are possible using resolution with additional variables,{{r|krishnamurthy}} or in stronger proof systems allowing the expression of additional constraints on local patterns of dominoes that can safely be avoided in any tiling and used to prune the search space.{{r|clausal}} Higher-level [[proof assistant]]s are capable of handling the coloring-based impossibility proof directly; these include [[Isabelle (proof assistant)|Isabelle]],{{r|isabelle}} the [[Mizar system]],{{r|mizar}} and [[Nqthm]].{{r|subramanian}}
In 1964, [[John McCarthy (computer scientist)|John McCarthy]] proposed the mutilated chessboard as a hard problem for [[automated proof]] systems, formulating it in [[first-order logic]] and asking for a system that can automatically determine the unsolvability of this formulation.{{r|mccarthy64}} Most considerations of this problem provide solutions "in the conceptual sense" that do not apply to McCarthy's logic formulation of the problem.{{r|andrews-bishop}} Despite the existence of general methods such as those based on graph matching, finding a solution to a logical formulation of the mutilated chessboard problem using the [[Resolution (logic)|resolution]] system of inference is exponentially hard,{{r|dantchev-riis|alekhnovich|razborov}} highlighting the need for methods in [[artificial intelligence]] that can automatically change to a more suitable problem representation{{r|korf}} and for [[knowledge representation]] systems that can manage the equivalences between different representations.{{r|kerber-pollet}} Short proofs are possible using resolution with additional variables,{{r|krishnamurthy}} or in stronger proof systems allowing the expression of additional constraints on local patterns of dominoes that can safely be avoided in any tiling and used to prune the search space.{{r|clausal}} Higher-level [[proof assistant]]s are capable of handling the coloring-based impossibility proof directly; these include [[Isabelle (proof assistant)|Isabelle]],{{r|isabelle}} the [[Mizar system]],{{r|mizar}} and [[Nqthm]].{{r|subramanian}}
{{-}}
{{-}}


Line 326: Line 326:
| volume = 22
| volume = 22
| year = 2014}}</ref>
| year = 2014}}</ref>

<ref name=kerber-pollet>{{citation
| last1 = Kerber | first1 = Manfred
| last2 = Pollet | first2 = Martin
| editor-last = Kohlhase | editor-first = Michael
| contribution = A tough nut for mathematical knowledge management
| contribution-url = https://www.cs.bham.ac.uk/~mmk/papers/05-MKM.html
| doi = 10.1007/11618027_6
| pages = 81–95
| publisher = Springer
| series = Lecture Notes in Computer Science
| title = Mathematical Knowledge Management, 4th International Conference, MKM 2005, Bremen, Germany, July 15-17, 2005, Revised Selected Papers
| volume = 3863
| year = 2005}}</ref>


<ref name=korf>{{citation
<ref name=korf>{{citation

Revision as of 01:13, 17 September 2022

The mutilated chessboard
Unsuccessful solution to the mutilated chessboard problem: as well as the two corners, two center squares remain uncovered

The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks:

Suppose a standard 8×8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares?

It is an impossible puzzle: there is no domino tiling 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, dominoes can cover all but two squares of the chessboard, if and only if the two uncovered squares are of different colors. This problem has been used as a test case for automated reasoning, creativity, and the philosophy of mathematics.

History

The mutilated chessboard problem is an instance of domino tiling of grids and polyominoes, also known as "dimer models", a general class of problems whose study in statistical mechanics dates to the work of Ralph H. Fowler and George Stanley Rushbrooke in 1937.[1] Domino tilings also have a long history of practical use in pavement design and the arrangement of tatami flooring.[2]

The mutilated chessboard problem itself was proposed by philosopher Max Black in his book Critical Thinking (1946), with a hint at the coloring-based solution to its impossibility.[3][4] It was popularized in the 1950s through later discussions by Solomon W. Golomb (1954),[5] George Gamow and Marvin Stern (1958),[6] Claude Berge (1958),[4][7] and Martin Gardner in his Scientific American column "Mathematical Games" (1957).[8]

The use of the mutilated chessboard problem in automated reasoning stems from a proposal for its use by John McCarthy in 1964.[9][10] It has also been studied in cognitive science as a test case for creative insight,[11][12][13] Black's original motivation for the problem.[3] In the philosophy of mathematics, it has been examined in studies of the nature of mathematical proof.[14][15][16][17]

Solution

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 both white. If the two white corners are removed from the board then 30 white squares and 32 black squares remain to be covered by dominoes, an impossibility. The case when the two black corners are removed is impossible by the same reasoning.[18] The same idea shows that no domino tiling can exist whenever any two squares of the same color (not just the opposite corners) are removed from the chessboard.[19]

Several other proofs of impossibility have also been found. A proof by Shmuel Winograd 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 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.[18][20] 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.[21]

Gomory's theorem: Removing any two oppositely-colored squares of a chessboard leaves a region that can be tiled by dominos. The two removed squares partition a Hamiltonian cycle through the squares into one (left) or two (right) paths through an even number of squares, allowing the modified chessboard to be tiled by dominos laid along the paths.
A region of the chessboard that has no domino tiling, but for which coloring-based impossibility proofs do not work

If two squares of opposite colors are removed, then the remaining board can always be tiled with dominoes; this result is Gomory's theorem,[22] after mathematician Ralph E. Gomory, whose proof was published in 1973.[19][20] Gomory's theorem can be proven using a Hamiltonian cycle of the grid graph 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.[22] Gomory's theorem is specific to the removal of only one square of each 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.[23]

Application to automated reasoning

Domino tiling problems on polyominoes, such as the mutilated chessboard problem, can be solved in polynomial time, either by converting them into problems in group theory[21][24] or as instances of bipartite matching. In the latter formulation, one obtains a bipartite graph 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 vertices of one color than the other implies that it fails the necessary conditions of Hall's marriage theorem, so no matching exists.[23][25][26] The problem can also be solved by formulating it as a constraint satisfaction problem, and applying semidefinite programming to a relaxation.[27]

In 1964, John McCarthy proposed the mutilated chessboard as a hard problem for automated proof systems, formulating it in first-order logic and asking for a system that can automatically determine the unsolvability of this formulation.[9] Most considerations of this problem provide solutions "in the conceptual sense" that do not apply to McCarthy's logic formulation of the problem.[28] Despite the existence of general methods such as those based on graph matching, finding a solution to a logical formulation of the mutilated chessboard problem using the resolution system of inference is exponentially hard,[29][30][31] highlighting the need for methods in artificial intelligence that can automatically change to a more suitable problem representation[32] and for knowledge representation systems that can manage the equivalences between different representations.[10] Short proofs are possible using resolution with additional variables,[33] or in stronger proof systems allowing the expression of additional constraints on local patterns of dominoes that can safely be avoided in any tiling and used to prune the search space.[34] Higher-level proof assistants are capable of handling the coloring-based impossibility proof directly; these include Isabelle,[35] the Mizar system,[36] and Nqthm.[37]

a8 black upside-down rook
h1 white circle
Wazir's tour problem

A similar problem asks if a wazir starting at a corner square of an unmutilated chessboard can visit every square exactly once, and finish at the opposite corner square. The wazir is a fairy chess piece that can only move 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 square of any complete tour is black. However, the opposite corner square is white.[38] This sort of tour of a chessboard also forms the basis of a type of puzzle called Numbrix, 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 corne.[39]

De Bruijn's theorem concerns the impossibility of packing certain cuboids into a larger cuboid. For instance, it is impossible, according to this theorem, to fill a 6 × 6 × 6 box with 1 × 2 × 4 cuboids. The proof uses a similar chessboard-coloring argument to the mutilated chessboard problem.[40]

References

  1. ^ Fowler, R. H.; Rushbrooke, G. S. (1937), "An attempt to extend the statistical theory of perfect solutions", Transactions of the Faraday Society, 33: 1272, doi:10.1039/tf9373301272
  2. ^ Erickson, Alejandro; Ruskey, Frank; Schurch, Mark; Woodcock, Jennifer (2010), "Auspicious tatami mat arrangements", in Thai, My T.; Sahni, Sartaj (eds.), Computing and Combinatorics, 16th Annual International Conference, COCOON 2010, Nha Trang, Vietnam, July 19–21, 2010, Proceedings, Lecture Notes in Computer Science, vol. 6196, Springer, pp. 288–297, arXiv:1103.3309, doi:10.1007/978-3-642-14031-0_32, MR 2720105
  3. ^ a b Black, Max (1946), Critical Thinking: An Introduction To Logic And Scientific Method, Prentice Hall, p. 157, 433
  4. ^ a b Robinson, J. A. (1991), "Formal and Informal Proofs", in Boyer, Robert S. (ed.), Automated Reasoning: Essays in Honor of Woody Bledsoe, Automated Reasoning Series, vol. 1, Springer Netherlands, pp. 267–282, doi:10.1007/978-94-011-3488-0_13; see especially Section 13.1, "The mutilated chess board problem", pp. 271–274 Archived 2022-07-18 at the Wayback Machine.
  5. ^ Golomb, S. W. (1954), "Checker boards and polyominoes", The American Mathematical Monthly, 61: 675–682, doi:10.1080/00029890.1954.11988548, JSTOR 2307321, MR 0067055
  6. ^ Gamow, George; Stern, Marvin (1958), "Domino game", Puzzle-Math, Viking Press, pp. 87–90, ISBN 978-0-333-08637-7
  7. ^ Berge, Claude (1958), Théorie des graphes et ses applications (in French), Dunod, p. 176
  8. ^ Gardner, Martin (February 1957), "An assortment of maddening puzzles", Mathematical Games, Scientific American, 196 (2): 152–158, JSTOR 24941903; for solution, see Gardner, Martin (March 1957), "Some old and new versions of ticktacktoe, plus the answers to last month's puzzles", Mathematical Games, Scientific American, 196 (3): 160–168, JSTOR 24940785
  9. ^ a b McCarthy, John (July 17, 1964), A tough nut for proof procedures, Stanford AI Memo, vol. 16, archived from the original on May 16, 2021, retrieved July 18, 2022
  10. ^ a b Kerber, Manfred; Pollet, Martin (2005), "A tough nut for mathematical knowledge management", in Kohlhase, Michael (ed.), Mathematical Knowledge Management, 4th International Conference, MKM 2005, Bremen, Germany, July 15-17, 2005, Revised Selected Papers, Lecture Notes in Computer Science, vol. 3863, Springer, pp. 81–95, doi:10.1007/11618027_6
  11. ^ Kaplan, Craig A.; Simon, Herbert A. (July 1990), "In search of insight", Cognitive Psychology, 22 (3): 374–419, doi:10.1016/0010-0285(90)90008-r
  12. ^ Akin, Ömer; Akin, Cem (January 1998), "On the process of creativity in puzzles, inventions, and designs", Automation in Construction, 7 (2–3): 123–138, doi:10.1016/s0926-5805(97)00057-5
  13. ^ 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", Cognitive Science, 43 (8), doi:10.1111/cogs.12771
  14. ^ MacKenzie, Donald (2005), "Computing and the cultures of proving", Philosophical Transactions of the Royal Society, 363 (1835): 2335–2350, doi:10.1098/rsta.2005.1649, JSTOR 30039731, MR 2197653
  15. ^ Kerber, Manfred (2014), "A proof and some representations", in Wyatt, Jeremy L.; Petters, Dean D.; Hogg, David C. (eds.), From Animals to Robots and Back: Reflections on Hard Problems in the Study of Cognition, A Collection in Honour of Aaron Sloman, Cognitive Systems Monographs, vol. 22, Springer International Publishing, pp. 65–73, doi:10.1007/978-3-319-06614-1_5
  16. ^ Tanswell, Fenner (2015), "A problem with the dependence of informal proofs on formal proofs", Philosophia Mathematica, Series III, 23 (3): 295–310, doi:10.1093/philmat/nkv008, MR 3404036
  17. ^ Starikova, Irina; Van Bendegem, Jean Paul (2021), "Revisiting the mutilated chessboard or the many roles of a picture", Logique et Analyse, 64 (255): 289–312, doi:10.2143/LEA.255.0.3290192, MR 4396339, archived from the original on 2022-07-19, retrieved 2022-07-19
  18. ^ a b McCarthy, John (1999), "Creative Solutions to Problems", AISB Workshop on Artificial Intelligence and Creativity, archived from the original on 2018-10-23, retrieved 2007-04-27
  19. ^ a b Honsberger, R. (1973), "Gomory's theorem", Mathematical Gems I, Mathematical Association of America, pp. 65–67
  20. ^ a b Mendelsohn, N. S. (2004), "Tiling with dominoes", The College Mathematics Journal, 35 (2): 115–120, doi:10.2307/4146865, JSTOR 4146865. Mendelsohn credits the original publication of Gomory's theorem to Honsberger (1973).
  21. ^ a b Propp, Jim (2021), "Conway's influence on the study of random tilings", The Mathematical Intelligencer, 43 (2): 40–46, doi:10.1007/s00283-021-10089-3, MR 4278473
  22. ^ a b Watkins, John J. (2004), Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, pp. 12–14, ISBN 978-0-691-11503-0
  23. ^ a b Wright, Colin (2014), "The mutilated chess board (revisited)" (PDF), Recreational Mathematics Magazine (1): 4–9, MR 3323392, archived (PDF) from the original on 2022-08-08, retrieved 2022-07-18
  24. ^ Thurston, William P. (1990), "Conway's tiling groups", The American Mathematical Monthly, 97 (8): 757–773, doi:10.2307/2324578, JSTOR 2324578, MR 1072815
  25. ^ Urquhart, Alasdair (2003), "Resolution proofs of matching principles", Annals of Mathematics and Artificial Intelligence, 37 (3): 241–250, doi:10.1023/A:1021231610627, MR 1969128
  26. ^ Codel, Cayden R.; Reeves, Joseph E.; Heule, Marijn J. H.; Bryant, Randal E. (2021), "Bipartite perfect matching benchmarks" (PDF), in Balyo, Tomáš; Froleyks, Nils; Heule, Marijn; Iser, Markus; Järvisalo, Matti; Suda, Martin (eds.), Proceedings of SAT Competition 2021: Solver and Benchmark Descriptions, Department of Computer Science Report Series B, vol. B-2021-1, Helsinki: Department of Computer Science, University of Helsinki, pp. 52–53, hdl:10138/333647, archived (PDF) from the original on 2022-07-18, retrieved 2022-07-18
  27. ^ de Klerk, Etienne; Van Maaren, Hans; Warners, Joost P. (2000), "Relaxations of the satisfiability problem using semidefinite programming", Journal of Automated Reasoning, 24 (1–2): 37–65, doi:10.1023/A:1006362203438, MR 1750258, archived from the original on 2021-06-20, retrieved 2022-07-19
  28. ^ Andrews, Peter B.; Bishop, Matthew (1996), "On sets, types, fixed points, and checkerboards", in Miglioli, Pierangelo; Moscato, Ugo; Mundici, Daniele; Ornaghi, Mario (eds.), Theorem Proving with Analytic Tableaux and Related Methods, 5th International Workshop, TABLEAUX '96, Terrasini, Palermo, Italy, May 15–17, 1996, Proceedings, Lecture Notes in Computer Science, vol. 1071, Springer, pp. 1–15, doi:10.1007/3-540-61208-4_1, archived from the original on 2022-07-18, retrieved 2022-07-18, 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
  29. ^ Dantchev, Stefan S.; Riis, Søren (2001), "'Planar' tautologies hard for resolution", 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14–17 October 2001, Las Vegas, Nevada, USA, IEEE Computer Society, pp. 220–229, doi:10.1109/SFCS.2001.959896, archived from the original on 14 September 2022, retrieved 18 July 2022
  30. ^ Alekhnovich, Michael (2004), "Mutilated chessboard problem is exponentially hard for resolution", Theoretical Computer Science, 310 (1–3): 513–525, doi:10.1016/S0304-3975(03)00395-5, MR 2020358
  31. ^ Razborov, Alexander A. (2004), "Resolution lower bounds for perfect matching principles" (PDF), Journal of Computer and System Sciences, 69 (1): 3–27, doi:10.1016/j.jcss.2004.01.004, MR 2070797, archived (PDF) from the original on 2022-08-08, retrieved 2022-07-19
  32. ^ Korf, Richard E. (1980), "Toward a model of representation changes", Artificial Intelligence, 14 (1): 41–78, doi:10.1016/0004-3702(80)90033-8, MR 0587851
  33. ^ Krishnamurthy, Balakrishnan (1985), "Short proofs for tricky formulas", Acta Informatica, 22 (3): 253–275, doi:10.1007/BF00265682, MR 0806206
  34. ^ Heule, Marijn J. H.; Kiesl, Benjamin; Biere, Armin (2019), "Clausal proofs of mutilated chessboards", in Badger, Julia M.; Rozier, Kristin Yvonne (eds.), NASA Formal Methods – 11th International Symposium, NFM 2019, Houston, TX, USA, May 7–9, 2019, Proceedings, Lecture Notes in Computer Science, vol. 11460, Springer, pp. 204–210, doi:10.1007/978-3-030-20652-9_13
  35. ^ Paulson, Lawrence C. (2001), "A simple formalization and proof for the mutilated chess board" (PDF), Logic Journal of the IGPL, 9 (3): 475–485, doi:10.1093/jigpal/9.3.475, MR 1828741, archived (PDF) from the original on 2022-08-08, retrieved 2022-07-18
  36. ^ Rudnicki, Piotr (1996), The Mutilated Checkerboard Problem in the Lightweight Set Theory of Mizar, Technical Report, vol. TR96-09, University of Alberta Department of Computer Science, doi:10.7939/R3QV3C738, archived from the original on 2022-07-18, retrieved 2022-07-18
  37. ^ Subramanian, Sakthi (1996), "An interactive solution to the mutilated checkerboard problem", Journal of Logic and Computation, 6 (4): 573–598, doi:10.1093/logcom/6.4.573, MR 1406233
  38. ^ Bivens, Irl C.; Holshouser, Arthur L.; Klein, Benjamin G. (October 2008), "Wazir circuits on an obstructed chessboard", Mathematics Magazine, 81 (4): 276–284, doi:10.1080/0025570X.2008.11953562, JSTOR 27643123
  39. ^ Hanson, Mary Grace; Nash, David A. (Spring 2018), "Minimal and maximal Numbrix puzzles", Pi Mu Epsilon Journal, 14 (8): 505–514, arXiv:1706.09389
  40. ^ de Bruijn, N. G. (1969), "Filling boxes with bricks", The American Mathematical Monthly, 76 (1): 37–40, doi:10.2307/2316785, JSTOR 2316785, MR 0234841
Wikimedia Commons has media related to Mutilated chessboard problem.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mutilated_chessboard_problem&oldid=1110703633"