THE FIVE DOGS PUZZLE.
(
Chessboard Problems)
In 1863, C.F. de Jaenisch first discussed the "Five Queens Puzzle"--to
place five queens on the chessboard so that every square shall be
attacked or occupied--which was propounded by his friend, a "Mr. de R."
Jaenisch showed that if no queen may attack another there are ninety-one
different ways of placing the five queens, reversals and reflections not
counting as different. If the queens may attack one another, I have
recorded hundreds of ways, but it is not practicable to enumerate them
exactly.
The illustration is supposed to represent an arrangement of sixty-four
kennels. It will be seen that five kennels each contain a dog, and on
further examination it will be seen that every one of the sixty-four
kennels is in a straight line with at least one dog--either
horizontally, vertically, or diagonally. Take any kennel you like, and
you will find that you can draw a straight line to a dog in one or other
of the three ways mentioned. The puzzle is to replace the five dogs and
discover in just how many different ways they may be placed in five
kennels _in a straight row_, so that every kennel shall always be in
line with at least one dog. Reversals and reflections are here counted
as different.
Answer:
The diagrams show four fundamentally different solutions. In the case of
A we can reverse the order, so that the single dog is in the bottom row
and the other four shifted up two squares. Also we may use the next
column to the right and both of the two central horizontal rows. Thus A
gives 8 solutions. Then B may be reversed and placed in either diagonal,
giving 4 solutions. Similarly C will give 4 solutions. The line in D
being symmetrical, its reversal will not be different, but it may be
disposed in 4 different directions. We thus have in all 20 different
solutions.
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