392.—­THE PEBBLE GAME.

In the case of fifteen pebbles, the first player wins if he first takes two.  Then when he holds an odd number and leaves 1, 8, or 9 he wins, and when he holds an even number and leaves 4, 5, or 12 he also wins.  He can always do one or other of these things until the end of the game, and so defeat his opponent.  In the case of thirteen pebbles the first player must lose if his opponent plays correctly.  In fact, the only numbers with which the first player ought to lose are 5 and multiples of 8 added to 5, such as 13, 21, 29, etc.

393.—­THE TWO ROOKS.

The second player can always win, but to ensure his doing so he must always place his rook, at the start and on every subsequent move, on the same diagonal as his opponent’s rook.  He can then force his opponent into a corner and win.  Supposing the diagram to represent the positions of the rooks at the start, then, if Black played first, White might have placed his rook at A and won next move.  Any square on that diagonal from A to H will win, but the best play is always to restrict the moves of the opposing rook as much as possible.  If White played first, then Black should have placed his rook at B (F would not be so good, as it gives White more scope); then if White goes to C, Black moves to D; White to E, Black to F; White to G, Black to C; White to H, Black to I; and Black must win next move.  If at any time Black had failed to move on to the same diagonal as White, then White could take Black’s diagonal and win.

    r:  black rook
    R:  white rook

+-+-+-+-+-+-+-+-+
|r| | | | | | | |
+-+-+-+-+-+-+-+-+
| |A| | | | | | |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| | | | |B|D|F| |
+-+-+-+-+-+-+-+-+
| | | | | |R|C|E|
+-+-+-+-+-+-+-+-+
| | | | | | |I|G|
+-+-+-+-+-+-+-+-+
| | | | | | | |H|
+-+-+-+-+-+-+-+-+

THE TWO ROOKS.

394.—­PUSS IN THE CORNER.

No matter whether he plays first or second, the player A, who starts the game at 55, must win.  Assuming that B adopts the very best lines of play in order to prolong as much as possible his existence, A, if he has first move, can always on his 12th move capture B; and if he has the second move, A can always on his 14th move make the capture.  His point is always to get diagonally in line with his opponent, and by going to 33, if he has first move, he prevents B getting diagonally in line with himself.  Here are two good games.  The number in front of the hyphen is always A’s move; that after the hyphen is B’s:—­

33-8, 32-15, 31-22, 30-21, 29-14, 22-7, 15-6, 14-2, 7-3, 6-4, 11-, and A must capture on his next (12th) move, -13, 54-20, 53-27, 52-34, 51-41, 50-34, 42-27, 35-20, 28-13, 21-6, 14-2, 7-3, 6-4, 11-, and A must capture on his next (14th) move.

395.—­A WAR PUZZLE GAME.