THE CROSS OF CARDS.
(
Problems Concerning Games.)
In this case we use only nine cards--the ace to nine of diamonds. The
puzzle is to arrange them in the form of a cross, exactly in the way
shown in the illustration, so that the pips in the vertical bar and in
the horizontal bar add up alike. In the example given it will be found
that both directions add up 23. What I want to know is, how many
different ways are there of rearranging the cards in order to bring
about this result? It will be seen that, without affecting the solution,
we may exchange the 5 with the 6, the 5 with the 7, the 8 with the 3,
and so on. Also we may make the horizontal and the vertical bars change
places. But such obvious manipulations as these are not to be regarded
as different solutions. They are all mere variations of one fundamental
solution. Now, how many of these fundamentally different solutions are
there? The pips need not, of course, always add up 23.
Answer:
There are eighteen fundamental arrangements, as follows, where I only
give the numbers in the horizontal bar, since the remainder must
naturally fall into their places.
5 6 1 7 4 2 4 5 6 8
3 5 1 6 8 3 4 5 6 7
3 4 1 7 8 1 4 7 6 8
2 5 1 7 8 2 3 7 6 8
2 5 3 6 8 2 4 7 5 8
1 5 3 7 8 3 4 9 5 6
2 4 3 7 8 2 4 9 5 7
1 4 5 7 8 1 4 9 6 7
2 3 5 7 8 2 3 9 6 7
It will be noticed that there must always be an odd number in the
centre, that there are four ways each of adding up 23, 25, and 27, but
only three ways each of summing to 24 and 26.
Informational
