ation sums that have the same product, and this may be done in
many ways. For example, 7 x 658 and 14 x 329 contain all the digits
once, and the product in each case is the same--4,606. Now, it will be
seen that the sum of the digits in the product is 16, which is neither
the highest nor the lowest sum so obtainable. Can you find the solution
of the problem that gives the lowest possible sum of digits in the
common product? Also that which gives the highest possible sum?

84.--THE PIERROT'S PUZZLE.

[Illustration]

The Pierrot in the illustration is standing in a posture that represents
the sign of multiplication. He is indicating the peculiar fact that 15
multiplied by 93 produces exactly the same figures (1,395), differently
arranged. The puzzle is to take any four digits you like (all different)
and similarly arrange them so that the number formed on one side of the
Pierrot when multiplied by the number on the other side shall produce
the same figures. There are very few ways of doing it, and I shall give
all the cases possible. Can you find them all? You are allowed to put
two figures on each side of the Pierrot as in the example shown, or to
place a single figure on one side and three figures on the other. If we
only used three digits instead of four, the only possible ways are
these: 3 multiplied by 51 equals 153, and 6 multiplied by 21 equals 126.

85.--THE CAB NUMBERS.

A London policeman one night saw two cabs drive off in opposite
directions under suspicious circumstances. This officer was a
particularly careful and wide-awake man, and he took out his pocket-book
to make an entry of the numbers of the cabs, but discovered that he had
lost his pencil. Luckily, however, he found a small piece of chalk, with
which he marked the two numbers on the gateway of a wharf close by. When
he returned to the same spot on his beat he stood and looked again at
the numbers, and noticed this peculiarity, that all the nine digits (no
nought) were used and that no figure was repeated, but that if he
multiplied the two numbers together they again produced the nine digits,
all once, and once only. When one of the clerks arrived at the wharf in
the early morning, he observed the chalk marks and carefully rubbed them
out. As the policeman could not remember them, certain mathematicians
were then consulted as to whether there was any known method for
discovering all the pairs of numbers that have the peculiarity that the
off