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 reader find similar arrangements producing 3, 4, 5, 6, 7, 8, and 9 respectively?  Also, can he find the pairs of smallest possible numbers in each case?  Thus, 1 4 6 5 8 divided by 7 3 2 9 is just as correct for 2 as the other example we have given, but the numbers are higher.

89.—­ADDING THE DIGITS.

If I write the sum of money, L987, 5s. 41/2d., and add up the digits, they sum to 36.  No digit has thus been used a second time in the amount or addition.  This is the largest amount possible under the conditions.  Now find the smallest possible amount, pounds, shillings, pence, and farthings being all represented.  You need not use more of the nine digits than you choose, but no digit may be repeated throughout.  The nought is not allowed.

90.—­THE CENTURY PUZZLE.

Can you write 100 in the form of a mixed number, using all the nine digits once, and only once?  The late distinguished French mathematician, Edouard Lucas, found seven different ways of doing it, and expressed his doubts as to there being any other ways.  As a matter of fact there are just eleven ways and no more.  Here is one of them, 91+5742/638.  Nine of the other ways have similarly two figures in the integral part of the number, but the eleventh expression has only one figure there.  Can the reader find this last form?

91.—­MORE MIXED FRACTIONS.

When I first published my solution to the last puzzle, I was led to attempt the expression of all numbers in turn up to 100 by a mixed fraction containing all the nine digits.  Here are twelve numbers for the reader to try his hand at:  13, 14, 15, 16, 18, 20, 27, 36, 40, 69, 72, 94.  Use every one of the nine digits once, and only once, in every case.