Chess-Piece Arrangement Problems

Part 1: — Unguard Arrangements with Leapers.

As the title indicates, this issue is the first of several devoted to new and collected results on chess-piece arrangement problems. Many problems using one type of piece are more of a mathematical than a chess nature, whereas problems with multiple types of piece, which we will study in later Parts, begin to approach the domain of true chess problems. In order to fit in more data we show most small-board diagrams with small squares. In these a vacant cell is shown grey, and unguarded pieces are shown by white sqares, guarded pieces by black squares instead of by chess symbols. Only the most interesting patterns, particularly those showing symmetry, have been diagrammed, however some further results will be found in the Appendix.

There are two principal types of problems: Unguards and Coverings. Roughly speaking, in unguards we place the maximum number of pieces so none guards any other, whereas in coverings we place the minimum number of pieces so that all cells are guarded. A normal chess piece has a given mode of movement and captures in the same way, that is it moves to the cell occupied by its victim which is then removed from the board. This is in contrast to special pieces that have different move and capture (for instance pawns). A cell is said to be guarded by a given piece if that piece could capture on that cell were it occupied by an opposing piece. The simplest pieces, leapers and composite leapers, are considered first, in order of increasing move coordinates, in later Parts the simplest riders and their compounds with leapers and with other riders, and finally hoppers and more complex pieces are considered.

We confine our attention mainly to problems using square boards, although we will see that solving problems on larger square boards sometimes requires knowledge of the numbers of arrangements possible on smaller rectangular boards. So future developments should aim to include results on rectangular boards. The first task is simply to find a solution fitting the conditions, this is often difficult enough, however there is then the supplementary question of enumerating all possible solutions. For a board of side n we denote by P(n) the number of pieces of the given type that solve the problem, by T(n) the number of different solutions, and by G(n) the number of geometrically distinct solutions, not regarding rotations and reflections as different. Chequering of the board in these problems serves only to help visualise the diagonals and is usually ignored in counting the arrangements. On the 1×1 board of course P = G = T = 1 for all pieces. Where G = 1 and T = 1 we give only the value for P.

I use the following terms to describe the various types of symmetry possible in square patterns: asymmetric = occurring in 8 forms by rotation and reflection, axial = having one axis of symmetry (which may be lateral or diagonal) and thus 4 forms, biaxial = having two axes of symmetry (both lateral or both diagonal), and thus 2 forms, rotary = invariant to 180° rotation (but not to any other rotation), so having 4 forms, birotary = invariant to 90° rotation, thus having 2 forms, octonary = invariant to all rotations and reflections, having only 1 form.

An {r,s}-leaper is said to be free on a given board if it is able to get from one cell to any other in a series of leaps. It follows that r and s must have no common factor, and further one must be even and the other odd, and finally (on a square board) they must be < half the side-length of the board. For all leapers of this type on a sufficiently large board the unguard task is solved by chequering. However, more pieces, or extra solutions with the same number of pieces, become possible if the board side is less than or equal to twice the longer move-coordinate of the leaper. Several cases where non-chequered solutions are possible on boards twice the longer move coordinate have been pointed out by Tom Marlow, 7 September 2004, and are now included below.

Wazir: {0,1} mover. This is the simplest case. To place maximum wazirs in unguard on any board we simply chequer the board and put them all on the same colour cells: in the case of an odd board (square or rectangle) the maximum is achieved by using those cells of the same colour as the corner cells, but on an even board either colour will do. The maximum number of wazirs that can be placed in unguard in these cases is therefore (n^2)/2 when n is even and (n^2 + 1)/2 when n is odd. On a rectangle m by n the number is mn/2 or (mn + 1)/2 according as mn is even or odd. Thus on any board G = 1. We show chequerings on all rectangular boards up to 5×5. Note that when both sides are odd (shown red) T = 1, but when one or both sides are even T = 2. This is the basis for calculating the totals T for many of the longer leapers.

For the longer {r,s} free leapers with r < s we consider arrangements on boards with side < 2s:

Knight: {1,2} leaper. 2×2 (P = 4), 3×3 (P = 5, G = 2, T = 2), 4×4 (P = 8, G = 3, T = 6, the two non-chequered solutions were spotted by T. W. Marlow). For larger boards chequering is the only way.

Zebra: {2,3} leaper. 2×2 (P = 4), 3×3 (P = 9), 4×4 (P = 12), 5×5 (P = 13, G = 4, T = 4), 6×6 (P = 20, G = 1, T = 2, special case found by T. W. Marlow). Chequering on larger boards.

Giraffe: {1,4} leaper. 2×2 (P = 4), 3×3 (P = 9), 4×4 (P = 16), 5×5 (P = 17, G = 2, T = 2), 6×6 (P = 20, G = 1, biaxial, T = 2), 7×7 (P = 25, G = 2, T = 2, chequered solution not shown), 8×8 (P = 32, G = 2, T = 4, chequered solution not shown, non-chequered solution found by T. W. Marlow). Chequering is the only solution on larger boards.

Antelope: {3,4} leaper. 2×2 (P = 4), 3×3 (P = 9), 4×4 (P = 16), 5×5 (P = 21), 6×6 (P = 24), 7×7 (P = 25, G = 8, two the same as for the giraffe, T = 8), 8×8 (P = 36, G = 1, T = 2, special case found by T. W. Marlow). Chequering on larger boards.

The same principle of solution will work for any pieces that guard or attack only dark squares when standing on light squares, or vice versa, and which also satisfy the freedom condition. For instance the panda which moves like a rook but only to squares of opposite shade.

This class consists of leapers able, from any cell, to access half the cells on any sufficiently large board. This means that (a) both coordinates must be odd, so that they are confined to cells of one colour on the chequered board, and (b) as for free leapers their move-coordinates, r and s, must have no common factor.

Fers: {1,1}-mover. 2×2 (P = 2, G = 1, T = 4), 3×3 (P = 6, G = 1, T = 2), 4×4 (P = 8, G = 7, 1 biaxial, 1 birotary, 2 axial, 3 asymmetric, T = 36), 5×5 (P = 15, G = 1, T = 2), 6×6 (P = 18, G = 46, of which 10 are symmetric, all axial, T = 328?), 7×7 (P = 28, G = 1, T = 2), 8×8 (P = 32, G = 477?, of which 41 are symmetrical; 32 axial, 3 rotary, 3 biaxial, 3 birotary, T = 3640?).

2×2, 3×3, 4×4 (7 cases) and 5×5

6×6 the 10 symmetric solutions

7×7 and 8×8 the 3 biaxial and 3 birotary solutions

Camel: {1,3} leaper. Like the wazir this piece has just one geometrically distinct solution on any square board. On all boards of side 5 or more this is a pattern of alternating stripes. 2×2 (P = 4), 3×3 (P = 9), 4×4 (P = 10, G = 1, T = 4), 5×5 (P = 15, G = 1, T = 2), 6×6 (P = 18, G = 1, T = 4), 7×7 (P = 28, G = 1, T = 2), 8×8 (P = 32, G = 1, T = 4).

This class (which might be called bounders instead of leapers!) consists of leapers whose coordinates have a common factor k > 1, in which case they are restricted to k^2 separate domains (when r + s is odd) or 2(k^2) separate domains (when r + s is even) — though possibly less on smaller boards.

Dabbaba: {0,2} leaper. On a large board a dabbaba has access to a quarter of the board, consisting of the cells with coordinates (odd,odd), (odd,even), (even,odd) or (even,even). To place the maximum dabbabas in unguard on each of these quarters is equivalent to the problem of placing wazirs on the board formed by closing up the spaces between the cells, i.e. they have to be arranged alternately, in chequered fashion. When these four chequerings are combined they result in patterns of two types, either a chequering of alternate 2×2 blocks of cells or alternate stripes two diagonals wide. We show the 8×8 solutions as typical (each is made up of four 4×4 solutions). 2×2 (P = 4), 3×3 (P = 5, G = 2, T = 8), 4×4 (P = 8, G = 6, T = 16), 5×5 (P = 13, G = 2, T = 8), 6×6 (P = 20, five 2×2 blocks in quincunx is the only solution), 7×7 (P = 25, G = 2, T = 8), 8×8 (P = 32, G = 6 ways, T = 16).