In order that a number of sixpences may not be divisible into a number
of equal piles, it is necessary that the number should be a prime. If
the banker can bring about a prime number, he will win; and I will show
how he can always do this, whatever the customer may put in the box, and
that therefore the banker will win to a certainty. The banker must first
deposit forty sixpences, and then, no matter how many the customer may
add, he will desire the latter to transfer from the counter the square
of the number next below what the customer put in. Thus, banker puts 40,
customer, we will say, adds 6, then transfers from the counter 25 (the
square of 5), which leaves 71 in all, a prime number. Try again. Banker
puts 40, customer adds 12, then transfers 121 (the square of 11), as
desired, which leaves 173, a prime number. The key to the puzzle is the
curious fact that any number up to 39, if added to its square and the
sum increased by 41, makes a prime number. This was first discovered by
Euler, the great mathematician. It has been suggested that the banker
might desire the customer to transfer sufficient to raise the contents
of the box to a given number; but this would not only make the thing an
absurdity, but breaks the rule that neither knows what the other puts
in.