THE TWO PAWNS.
(
The Guarded Chessboard)
Here is a neat little puzzle in counting. In how many different ways may
the two pawns advance to the eighth square? You may move them in any
order you like to form a different sequence. For example, you may move
the Q R P (one or two squares) first, or the K R P first, or one pawn as
far as you like before touching the other. Any sequence is permissible,
only in this puzzle as soon as a pawn reaches the eighth square it is
dead, and remains there unconverted. Can you count the number of
different sequences? At first it will strike you as being very
difficult, but I will show that it is really quite simple when properly
attacked.
VARIOUS CHESS PUZZLES.
"Chesse-play is a good and wittie exercise of
the minde for some kinde of men."
Burton's _Anatomy of Melancholy_.
Answer:
Call one pawn A and the other B. Now, owing to that optional first move,
either pawn may make either 5 or 6 moves in reaching the eighth square.
There are, therefore, four cases to be considered: (1) A 6 moves and B 6
moves; (2) A 6 moves and B 5 moves; (3) A 5 moves and B 6 moves; (4) A 5
moves and B 5 moves. In case (1) there are 12 moves, and we may select
any 6 of these for A. Therefore 7x8x9x10x11x12 divided by 1x2x3x4x5x6
gives us the number of variations for this case--that is, 924. Similarly
for case (2), 6 selections out of 11 will be 462; in case (3), 5
selections out of 11 will also be 462; and in case (4), 5 selections out
of 10 will be 252. Add these four numbers together and we get 2,100,
which is the correct number of different ways in which the pawns may
advance under the conditions. (See No. 270, on p. 204.)
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