çArrangements
This page has been reproduced September 2010, from the online Games and Puzzles Journal
site with minimal alteration.
The Games and Puzzles
Journal — Issue 40, July-August 2005 |
Chess-Piece Arrangement Problems
by George Jelliss
Part 5: — The Eight Officers
Sections on this page: —
(1) Introduction —
(2) Unguard Arrangements —
(3) Coverings —
(4) Packings —
(5) Other problems —
End.
Having studied arrangement problems where the number of pieces can be varied, we now turn to some in which the set of pieces to be used is fixed.
The most popular set of pieces for arrangement problems consists of the eight back-row pieces of one colour from a chess set, that is excluding the pawns.
From a mathematical point of view the choice may be considered purely arbitrary, but they do all go together rather well to produce some interesting results.
It should be noted that the two bishops are normally assumed to run on cells of opposite colour in the chequering, but this rule may be ignored in exceptional cases.
T. R. Dawson, London Evening Standard 4 August 1930 gave a set of arrangements that he called “Brighton Beach”. The eight pieces, mutually unguarded, occupy the smallest rectangular area. Dawson gave six solutions on the 5×6 area:
T. R. Dawson, British Chess Magazine August 1941 also considered the smallest triangular area (half 7×7), giving three solutions:
Spanning. P. Frey, L'Echo de Paris 1926-7 found a position (A) in which the eight officers span the chessboard; that is they are unguarded and guard all vacant cells. (This arrangement can be torally rotated, as if the left and right sides and the top and bottom sides of the board are connected, to give other similar solutions: a1 to b2, a1 to c2, a1 to c3).
— By making g7 = R+Kt and f3 = B+Kt we get a spanning arrangement with eight different officers (counting bishops on different colours as different types). The three Knight components can also be replaced by Nightriders.
Domination. It is impossible to arrange the eight officers to dominate the board, that is to guard all 64 cells.
The best they can do is 63 (see below). We can however, paradoxically, manage to guard all 64 cells by putting the two bishops on the same colour cells.
I gave the position (B) above in Chessics 1984, but it was probably known earlier. Does anyone have an earlier source?
— By using a Queen + Knight (Amazon) in place of the Queen, which is equivalent to putting the queen and knight on the same cell, we can guard all 64 cells, and furthermore the second knight is then not needed. This arrangement is due to M. Keller World Game Review #10: Kc2, A(Q+Kt)e4, Ra8, Rh7, Bd3, Bf2.
Near-Domination. In the following 20 arrangements only one of the 64 cells is unguarded. In the first 12 it is occupied by one of the pieces (therefore shown white).
The first eight results were given by T. R. Dawson in L'Echiquier January and February 1932. The sources of the others are:
(9) T. R. Dawson, British Chess Magazine July 1938; (10) W. H. Paul Fairy Chess Review 1939; (11) S. G. Liddell FCR 1939; (12) W. Cross and A. H. Haddy FCR 1940.
In the next 8 positions, the unguarded cell is vacant, or can be occupied, as shown here, by the White King in stalemate.
Positions (13)-(18) were given by T. R. Dawson L'Echiquier January-February 1932. Position (19) T. R. Dawson, British Chess Magazine August 1938. Position (20), where the Ke7 also be at f7, G. P. Jelliss 1968. (If I remember correctly this was circulated to the Fairy Chess Correspondence Circle.
It was presented as a Stalemate in 2, with white pieces and black king. The rooks g1, h2 starting at g8, f2, king e7 at f7, and b3 replaced by a "Flying King" d1. The Flying King can move to any unguarded cell and thus capture any unguarded piece. Solution: 1.Rh2, Kc1/e1/f1 2.Rg1+ Kb3 3.Ke7 stalemate. If 1...Kb3 then 2.Rg1 stalemate.)
The first ever Eight Officers construction task (A) was to arrange them on the board so as to maximise their "freedom of movement" and shows 100 moves (M. Bezzel, Schachzeitung 1848), one piece is guarded, and there are 7 unguarded vacant cells.
With any given set of pieces we can set the packing problem: to guard as few cells as possible. We can call the solution loose if all the pieces are unguarded and close if they are all guarded; between these we can have mixed solutions.
The loose packing problem for the eight officers on the chessboard left unguarded was considered in British Chess Magazine 1968, subject to the Black King being included and in stalemate, and E. Fielder gave position (B) with 17 squares unguarded and Black King in stalemate at h7. The cells unguarded are a1237,b1234,c1234,d1234,h7.
Without the extra condition 18 unguarded cells can be found as in (C), by me in Chessics 1984; there is nowhere for Black King in stalemate. The cells unguarded are a1234,b1234,c1234,d234,f128.
Diagram (D) arranges the pieces so as to guard as few vacant cells, 8, as possible (W. A. Shinkman, source?).
The eight officers of each colour appear at the start of a game all lined up along their first rank.
Various proposals have been made from time to time for arranging them in some other order than the traditional one.
The number of ways they could be arranged is well known to be 2880 (equal to 4×4×6×5×4×3/2 the product of the choices of placing B, B, Q, K, R, R, successively).
Various of these one-line arrangements have special properties, some are presented in the following problems, which are phrased so as to have unique solutions.
Place the eight white officers along the first rank, with Q to left of K, so that:
(1) all men are guarded:
or: and similar.
(2) all men are guarded except the king:
or: and similar.
(3) all men are guarded if the board is cylindrical but not if it is as normal:
(rather familiar!).
Could it be that chess was originally invented for play on a cylindrical or circular board?!
(4) so that as many cells as possible in the white half of the board are guarded:
or: guard 29/32 cells.
(5) so that as many cells as possible in the whole board are guarded:
guards 45/64 cells (sent to me by Ivor Sanders June 1991).
To Be Continued (Offers of any other constructions with the eight officers will be welcome).
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