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.