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(Money Puzzles)
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
officer had noticed; but they knew of none. The investigation, however,
was interesting, and the following question out of many was proposed:
What two numbers, containing together all the nine digits, will, when
multiplied together, produce another number (the _highest possible_)
containing also all the nine digits? The nought is not allowed anywhere.

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