MISCELLANEOUS PUZZLES
73.—
The Game of Kayles.
To win at this game you must, sooner or later, leave your
opponent an even number of similar groups. Then whatever he
does in one group you repeat in a similar group. Suppose, for
example, that you leave him these groups : 0 . 0 . 000 . 000.
Now, if he knocks down a single, you knock down a single ; if he
knocks down two in one triplet, you knock down two in the other
triplet; if he knocks down the central kayle in a triplet, you knock
down the central one in the other triplet. In this way you
must eventually win. As the game is started with the arrangement
0 . 00000000000, the first player can always win, but only by
knocking down the sixth or tenth kayle (counting the one already
fallen as the second), and this leaves in either case 0 . 000 . 0000000,
as the order of the groups is of no importance. Whatever the
second player now does, this can always be resolved into an even
number of equal groups. Let us suppose that he knocks down the
single one, then we play to leave him 00 . 0000000. Now, what-
ever he does we can afterwards leave him either 000 . 000 or
0 . 00 . 000. We know why the former wins, and the latter
wins also, because, however he
may play, we can always leave
him either 0.0, or 0.0.0.0,
or 00 . 00, as the case may
be. The complete analysis I
can now leave for the amuse-
ment of the reader.
74.—
The Broken Chessboard.
The illustration will show
how the thirteen pieces can
be put together so as to con-
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