Part 42

Where n = 5 there are 2 fundamental solutions and 10 in all.

Where n = 6 there is 1 fundamental solution and 4 in all.

Where n = 7 there are 6 fundamental solutions and 40 in all.

Where n = 8 there are 12 fundamental solutions and 92 in all.

Where n = 9 there are 46 fundamental solutions.

Where n = 10 there are 92 fundamental solutions.

Where n = 11 there are 341 fundamental solutions.

Obviously n rooks may be placed without attack on an n squared board in n! ways, but how many of these are fundamentally different I have only worked out in the four cases where n equals 2, 3, 4, and 5. The answers here are respectively 1, 2, 7, and 23. (See No. 296, "The Four Lions.")

We can place 2n-2 bishops on an n squared board in 2^{n} ways. (See No. 299, "Bishops in Convocation.") For boards containing 2, 3, 4, 5, 6, 7, 8 squares, on a side there are respectively 1, 2, 3, 6, 10, 20, 36 fundamentally different arrangements. Where n is odd there are 2^{1/2(n-1)} such arrangements, each giving 4 by reversals and reflections, and 2^{n-3} - 2^{1/2(n-3)} giving 8. Where n is even there are 2^{1/2(n-2)}, each giving 4 by reversals and reflections, and 2^{n-3} - 2^{1/2(n-4)}, each giving 8.

We can place 1/2(n squared+1) knights on an n squared board without attack, when n is odd, in 1 fundamental way; and 1/2n squared knights on an n squared board, when n is even, in 1 fundamental way. In the first case we place all the knights on the same colour as the central square; in the second case we place them all on black, or all on white, squares.

THE TWO PIECES PROBLEM.

On a board of n squared squares, two queens, two rooks, two bishops, or two knights can always be placed, irrespective of attack or not, in 1/2(n^{4} - n squared) ways. The following formulae will show in how many of these ways the two pieces may be placed with attack and without:--

With Attack. Without Attack.

2 Queens 5n cubed - 6n squared + n 3n^{4} - 10n cubed + 9n squared - 2n ------------------- ------------------------------ 3 6

2 Rooks n cubed - n squared n^{4} - 2n cubed + n squared ---------------------- 2

2 Bishops 4n cubed - 6n squared + 2n 3n^{4} - 4n cubed + 3n squared - 2n -------------------- ----------------------------- 6 6

2 Knights 4n squared - 12n + 8 n^{4} - 9n squared + 24n -------------------- 2

(See No. 318, " Lion Hunting.")

DYNAMICAL CHESS PUZZLES.

"Push on--keep moving." THOS. MORTON: Cure for the Heartache.

320.--THE ROOK'S TOUR.

+---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | R | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+

The puzzle is to move the single rook over the whole board, so that it shall visit every square of the board once, and only once, and end its tour on the square from which it starts. You have to do this in as few moves as possible, and unless you are very careful you will take just one move too many. Of course, a square is regarded equally as "visited" whether you merely pass over it or make it a stopping-place, and we will not quibble over the point whether the original square is actually visited twice. We will assume that it is not.

321.--THE ROOK'S JOURNEY.

This puzzle I call "The Rook's Journey," because the word "tour" (derived from a turner's wheel) implies that we return to the point from which we set out, and we do not do this in the present case. We should not be satisfied with a personally conducted holiday tour that ended by leaving us, say, in the middle of the Sahara. The rook here makes twenty-one moves, in the course of which journey it visits every square of the board once and only once, stopping at the square marked 10 at the end of its tenth move, and ending at the square marked 21. Two consecutive moves cannot be made in the same direction--that is to say, you must make a turn after every move.

+---+---+---+---+---+---+---+---+ | | | | | | | | R | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | 21| | 10| | | | | +---+---+---+---+---+---+---+---+

322.--THE LANGUISHING MAIDEN.

--+-----+-----+-----+-----+-----+-----+-----+ | | | | | | | | | Kt | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | M | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+