The Games and Puzzles Journal — Issue 39, May-June 2005

Chess-Piece Arrangement Problems

by George Jelliss

Part 4: — Multiple Unguard Arrangements

Back to: GPJ Index Page Sections on this page: (1) Introduction. (2) Two Types. (3) Three Types. (4) Four of Five Types. (5) Other Multiple Arrangements. End.
(1) Introduction é
Problems with multiple types of piece begin to approach the domain of true chess problems. Unguard problems can be extended to include arrangements in which a mixture of two or more types occupies the board. This recreation was introduced by me in Chessics #13 (1982). The general problem is to place a pieces of type A, b pieces of type B, c pieces of type C, and so on, on the board, none guarding any other, so that the product a×b×c×... of the numbers of pieces of each type is maximised.

There must be at least one piece of each type specified, since if there are no pieces of type A, say, the product is minimised (a = 0 and so the product is 0). Usually at least two pieces of each type are required, the reason being that an increase from 1 to 2 doubles the product, whereas an increase from 2 to 3 only multiplies it by 1.5. And, generally, an increase from k to k+1 only multiplies it by (1 + 1/k) which gets smaller as k increases. An exception to this rule occurs when there are so many types being used that there is just no room for more than one of the larger pieces.

Another principle that is sometimes useful in constructing these arrays is that if there are n spaces to be distributed between two types of piece then, to get the largest product, they should be shared out as near as possible equally. This is because for any number k we always have k^2 = k×k > (k–1)×(k+1) = k^2 – 1. This applies for example in the QR construction, where we can have no more than 8 pieces on the board (one in each rank and file) and for maximum q×r we must have 4 of each (4×4 = 16, not 3×5 = 15).

(2) Multiple Unguard Arrangements: two types of piece. é

Here are the best results obtained using the orthodox chess-pieces, beginning with two types of piece.
(KQ: 10×4 = 40)
(KR: 8×5 = 40)
(KB: 9×8 = 72)
(KS: 10×14 = 140)
(QR: 4×4 = 16)
(QB: 5×6 = 30)
(QS: 2×15 = 30)
(RB: 5×7 = 35)
(RS: 4×16 = 64)
(BS: 7×16 = 112)
The results for KQ, KR, QR, QB, QS, RS, BS are from my original article. (However I have recently noticed that the 5Q + 6B arrangement was given, as an isolated puzzle, by V. Onitiu in Problemist Fairy Chess Supplement 1931, problem 313). The result for KB is due to W. H. Duce. The results for KS, RB are due to S. Ylikarjula.

(3) Multiple Unguard Arrangements: three types of piece. é

Results for three types:
(KQR: 8×2×3 = 48)
(KQB: 6×3×5 = 90)
(KQS: 8×2×8 = 128)
(KRB: 5×4×5 = 100)
(KRS: 8×4×8 = 256)
(KBS: 8×7×9 = 504)
(QRB: 3×3×4 = 36)
(QRS: 2×3×8 = 48)
(QBS: 2×6×7 = 84)
(RBS: 3×7×10 = 210)
The result for QRB is the only survivor from my original article. The result for KRS is due to W. H. Duce. The rest are due to S. Ylikarjula.

(4) Multiple Unguard Arrangements: four and five types of piece. é

Results for four and five types:
(KQRB: 5×2×2×4 = 80)
(KQRS: 7×1×3×8 = 168)
(KQBS: 7×2×4×5 = 280)
(KRBS: 6×2×6×8 = 576)
(QRBS: 1×3×4×9 = 108)
(KQRBS: 4×1×3×3×8 = 288)
The KRBS result is due to Andy Sag and was sent to me from Australia by Arthur Willmott in 1993. The other results are all due to Simo Ylikarjula of Finland and appeared in Chessics #16 in 1983.

(5) Other Multiple Arrangements é

Are there other functions of the numbers a, b, c, ... that might give interesting results? The sum of squares for example? I don't see why the multiplication of the numbers should be especially significant, other than being the simplest formula. This may be a line for further research.

There are certainly other interesting positions that are excluded if we examine only the cases where the product is maximum. For example there is just one way, apart from rotations and reflections, of placing 6 queens and 4 bishops in unguard on the chessboard, as shown here:
The 5Q + 6B and the 10K + 4Q solutions are also unique.

The same question can be asked concerning coverings of the board by mixtures of pieces. This arrangement of 3Q + 2R to dominate the board is given by C. Planck in Chess Amateur 1908, p.305.

Further examples may be added. Any new results will be welcome.

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