icer 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.

86.--QUEER MULTIPLICATION.

If I multiply 51,249,876 by 3 (thus using all the nine digits once, and
once only), I get 153,749,628 (which again contains all the nine digits
once). Similarly, if I multiply 16,583,742 by 9 the result is
149,253,678, where in each case all the nine digits are used. Now, take
6 as your multiplier and try to arrange the remaining eight digits so as
to produce by multiplication a number containing all nine once, and once
only. You will find it far from easy, but it can be done.

87.--THE NUMBER-CHECKS PUZZLE.

[Illustration]

Where a large number of workmen are employed on a building it is
customary to provide every man with a little disc bearing his number.
These are hung on a board by the men as they arrive, and serve as a
check on punctuality. Now, I once noticed a foreman remove a number of
these checks from his board and place them on a split-ring which he
carried in his pocket. This at once gave me the idea for a good puzzle.
In fact, I will confide to my readers that this is just how ideas for
puzzles arise. You cannot really create an idea: it happens--and you
have to be on the alert to seize it when it does so happen.

It will be seen from the illustration that there are ten of these
checks on a ring, numbered 1 to 9 and 0. The puzzle is to divide them
into three groups without taking any off the ring, so that the first
group multiplied by the second makes the third group. For example, we
can divide them into the three groups, 2--8 9 0 7--1 5 4 6 3, by
bringing the 6 and the 3 round to the 4, but unfortunately the first
two when multiplied together do not make the third. Can you separate
them correctly? Of course you may have as many of the checks as you
like in any group. The puzzle calls for some ingenuity, unless you
have the luck to hit on the answer by chance.

88.--DIGITAL DIVISION.

It is another good puzzle so to arrange the nine digits (the nought
excluded) into two groups so that one group when divided by the other
produces a given number without remainder. For example, 1 3 4 5 8
divided by 6 7 2 9 gives 2. Can the