CARD TRIANGLES.
(
Problems Concerning Games.)
Here you pick out the nine cards, ace to nine of diamonds, and arrange
them in the form of a triangle, exactly as shown in the illustration, so
that the pips add up the same on the three sides. In the example given
it will be seen that they sum to 20 on each side, but the particular
number is of no importance so long as it is the same on all three sides.
The puzzle is to find out in just how many different ways this can be
done.
If you simply turn the cards round so that one of the other two sides is
nearest to you this will not count as different, for the order will be
the same. Also, if you make the 4, 9, 5 change places with the 7, 3, 8,
and at the same time exchange the 1 and the 6, it will not be different.
But if you only change the 1 and the 6 it will be different, because the
order round the triangle is not the same. This explanation will prevent
any doubt arising as to the conditions.
Answer:
The following arrangements of the cards show (1) the smallest possible
sum, 17; and (2) the largest possible, 23.
1 7
9 6 4 2
4 8 3 6
3 7 5 2 9 5 1 8
It will be seen that the two cards in the middle of any side may always
be interchanged without affecting the conditions. Thus there are eight
ways of presenting every fundamental arrangement. The number of
fundamentals is eighteen, as follows: two summing to 17, four summing to
19, six summing to 20, four summing to 21, and two summing to 23. These
eighteen fundamentals, multiplied by eight (for the reason stated
above), give 144 as the total number of different ways of placing the
cards.
