The answer is that there are just 2,400 different ways. Any three coins
may be taken from one side to combine with one coin taken from the other
side. I give four examples on this and the next page. We may thus select
three from the top in ten ways and one from the bottom in five ways,
making fifty. But we may also select three from the bottom and one from
the top in fifty ways. We may thus select the four coins in one hundred
ways, and the four removed may be arranged by permutation in twenty-four
ways. Thus there are 24 × 100 = 2,400 different solutions.
[Illustration]
As all the points and lines puzzles that I have given so far, excepting
the last, are variations of the case of ten points arranged to form five
lines of four, it will be well to consider this particular case
generally. There are six fundamental solutions, and no more, as shown in
the six diagrams. These, for the sake of convenience, I named some years
ago the Star, the Dart, the Compasses, the Funnel, the Scissors, and the
Nail. (See next page.) Readers will understand that any one of these
forms may be distorted in an infinite number of different ways without
destroying its real character.
In "The King and the Castles" we have the Star, and its solution gives
the Compasses. In the "Cherries and Plums" solution we find that the
Cherries represent the Funnel and the Plums the Dart. The solution of
the "Plantation Puzzle" is an example of the Dart distorted. Any
solution to the "Ten Coins" will represent the Scissors. Thus examples
of all have been given except the Nail.
On a reduced chessboard, 7 by 7, we may place the ten pawns in just
three different ways, but they must all represent the Dart. The
"Plantation" shows one way, the Plums show a second way, and the reader
may like to find the third way for himself. On an ordinary chessboard, 8
by 8, we can also get in a beautiful example of the Funnel--symmetrical
in relation to the diagonal of the board. The smallest board that will
take a Star is one 9 by 7. The Nail requires a board 11 by 7, the
Scissors
[Illustration]
11 by 9, and the Compasses 17 by 12. At least these are the best results
recorded in my note-book. They may be beaten, but I do not think so. If
you divide a chessboard into two parts by a diagonal zigzag line, so
that the larger part contains 36 squares and the smaller part 28
squares, you can place three separate schemes on the larger part and one
on the smaller part (all Darts) without their conflicting--that is, they
occupy forty different squares. They can be placed in other ways without
a division of the board. The smallest square board that will contain six
different schemes (not fundamentally different), without any line of one
scheme crossing the line of another, is 14 by 14; and the smallest board
that will contain one scheme entirely enclosed within the lines of a
second scheme, without any of the lines of the one, when drawn from
point to point, crossing a line of the other, is 14 by 12.
[Illustration: STAR DART COMPASSES FUNNEL SCISSORS NAIL]