If we remove the ace, the remaining cards may he divided into two groups
(each adding up alike) in four ways; if we remove 3, there are three
ways; if 5, there are four ways; if 7, there are three ways; and if we
remove 9, there are four ways of making two equal groups. There are thus
eighteen different ways of grouping, and if we take any one of these and
keep the odd card (that I have called "removed") at the head of the
column, then one set of numbers can be varied in order in twenty-four
ways in the column and the other four twenty-four ways in the
horizontal, or together they may be varied in 24 × 24 = 576 ways. And as
there are eighteen such cases, we multiply this number by 18 and get
10,368, the correct number of ways of placing the cards. As this number
includes the reflections, we must divide by 2, but we have also to
remember that every horizontal row can change places with a vertical
row, necessitating our multiplying by 2; so one operation cancels the
other.