[Illustration: 1]
[Illustration: 2]
[Illustration: 3]
[Illustration: 4]
Let us use the letters A, K, Q, J, to denote ace, king, queen, jack; and
D, S, H, C, to denote diamonds, spades, hearts, clubs. In Diagrams 1
and 2 we have the two available ways of arranging either group of
letters so that no two similar letters shall be in line--though a
quarter-turn of 1 will give us the arrangement in 2. If we superimpose
or combine these two squares, we get the arrangement of Diagram 3, which
is one solution. But in each square we may put the letters in the top
line in twenty-four different ways without altering the scheme of
arrangement. Thus, in Diagram 4 the S's are similarly placed to the D's
in 2, the H's to the S's, the C's to the H's, and the D's to the C's. It
clearly follows that there must be 24×24 = 576 ways of combining the two
primitive arrangements. But the error that Labosne fell into was that of
assuming that the A, K, Q, J must be arranged in the form 1, and the D,
S, H, C in the form 2. He thus included reflections and half-turns, but
not quarter-turns. They may obviously be interchanged. So that the
correct answer is 2 × 576 = 1,152, counting reflections and reversals as
different. Put in another manner, the pairs in the top row may be
written in 16 × 9 × 4 × 1 = 576 different ways, and the square then
completed in 2 ways, making 1,152 ways in all.