ç Index to Chessboard Dissection Problems

Chessboard Dissection Problems

These pages are devoted to problems involving shapes formed by joining squares together edge to edge, and patterns formed by fitting such shapes together, jig-saw fashion, or by cutting them out of larger arrays of squares, such as chessboards.

Historical Introduction

Commercially produced puzzles consisting of pieces of various numbers of squares that have to be put together to make a small facsimile of a chessboard are probably the origin of this subject, although of course geometrical dissections, such as proofs of the Pythagorean theorem, involving more general shapes, are much older. The earliest example with a definite date is a US patent taken out by Henry Luers in 1880 reported by Jerry Slocum in his Compendium of Checkerboard Puzzles (1983). This consisted of 15 pieces of 2, 3, 4 and 5 cells.

A more systematic mathematical approach was adopted by H. E. Dudeney in two of his puzzles. One in the magazine Tribune 1906 requires a chessboard to be broken up into as many pieces as possible, all of different shape or colouring. This uses all shapes of 1 to 4 squares, plus a 1×8 piece. The same problem appeared also in Sam Loyd's Puzzle Magazine (April-July 1908) and in Dudeney's Amusements in Mathematics 1917, problem 293. (This problem is often attributed to Loyd, who may have published it earlier in some other periodical.) The other Dudeney pioneer is problem 74 in The Canterbury Puzzles and Other Curious Problems, published in 1907. This requires 13 pieces of a broken chessboard to be put together again and uses all 12 different 5-square shapes plus a 2×2 square.

Some results related to these two problems were sent by F. Douglas, H. D. Benjamin, A. M. Holmes and R. F. Smith to The British Chess Magazine October 1933 and June 1934.

Much pioneering work on the subject, using all the pieces from 1 to 6 squares, was done in the pages of The Problemist Fairy Chess Supplement (1930-6) and its continuation The Fairy Chess Review (1936-1958), during the years 1934-57. This work, associated with the names of H. D. Benjamin, T. R. Dawson, F. Hansson, W. E. Lester, W. Stead, and many others has not been fully appreciated because the results were published there in coded form instead of by means of diagrams. This was due in part to the cost of making blocks for the letterpress printing method then in use. In 1986 (as reported in Chessics #28) I decoded most of the results in FCR. The results occupied 63 A4 pages, which in 1988 I drew out in reasonably tidy form and copied, as Dissection Problems in PFCS/FCR, to a few correspondents interested in the subject. With modern methods of graphics and it is now possible to present these results visually in these pages.

The first book on the subject was Polyominoes (1965) by S. W. Golomb. The term 'polyomino' for any piece composed of squares joined edge-to-edge having been coined by him in a Harvard lecture given in 1953, on the false analogy (as linguists call it) that a 'domino', formed of two squares, is a contraction of 'di-omino'). The shorter form 'ominoes' has also become common and for each value of n (1, 2, 3, ...) there is a corresponding name for n-square pieces using a numerical prefix of Greek derivation: 'monomino', 'domino', 'tromino', 'tetromino', 'pentomino', 'hexomino', 'heptomino', and so on. Personally I dislike these superfluous terms and avoid them here, since I consider the description 'n-square pieces' perfectly adequate.

A bibliography covering more recent work, compiled by W. L. Schaaf, appeared in the Journal of Recreational Mathematics (volume 16, issue 4, 1983-4).

Structural and Aesthetic Features

Various special features of dissection patterns can be detected and we define here some of the terms used to describe them. Different constructors place different emphasis on the importance or insignificance of these features. For purposes of a 'task' problem where the aim is to demonstrate a difficult dissection such incidental features can be ignored. However where more freedom of placing the pieces is available, solutions showing these special features are naturally preferred.

A fault line in a dissection is a straight line of shape-edges that runs right across the pattern. A dissection pattern is said to show crossroads if there is at least one point where four pieces meet, forming a lateral cross (+). Most dissections can be achieved under this extra constraint, and it is generally held to add to the artistic attractiveness of the pattern, as well as testing the ingenuity and patience of the constructor just a little more. However there are cases where it cannot be avoided, or where the number of crossroads may be maximised.

The pages of FCR offer no guidance to would-be constructors as to any systematic way of solving such abstract jig-saw puzzles as these. The only hint that I can offer is that it seems to help if the clumps, pieces incorporating a 2×2 part, are as far as possible reserved for later insertion, particularly the asymmetric clumps which have more possible positions.

A construction feature that is of interest in smaller arrangements is to distinguish between pieces that abut the outer edges of the pattern. For short we describe these as being on edge, pieces that do not abut the edge we call internal. In some cases it is helpful to show the outline dividing the internal and edge pieces by a bolder line.

In the case of holes in a shape the convention is adopted of showing them shaded in if they are small but left white if they are larger and cannot be confused with the construction pieces. Of course there is an ambiguity as to whether an area outlined by pieces represents a hole or another piece. Sometimes we count it as a piece and sometimes as a hole depending on what sets of pieces we are concentrating attention on. The same goes for indentations round the edges. We usually leave these white, but sometimes count them as holes or as pieces, or parts thereof. This ambiguity occurs, for instance, in the popular problem of placing the twelve 5-square pieces on the 8×8 chessboard leaving four squares uncovered.

One of the fascinating aspects of polyominoes is the curious mixture that they provide of arithmetic and geometry. To cover an area of N squares we must select an appropriate set of pieces, and preferably the choice should not be arbitrary, nor the result of too many conditions.

To show consistency in the diagrams in this study I have drawn most of the figures with the unit squares the same size, namely one eighth of an inch (0.75 pica) wide. This gives 48 units across an A4 page (between inch margins), sufficient to show five 8×8 boards, with unit spacing.

In formulas I show multiplication by a middot (·) but use the diagonal cross (×) as an abbreviation for 'by' in stating the size of rectangular boards (m×n). For example, a board of 30 cells may be 1×30 or 2×15 or 3×10 or 5×6.

In counting the number of solutions we consider only the number of generic or 'geometrically distinct' arrangements—rotations, reflections and chequerings of the patterns being discounted.