## Multiple Addition

EDMUND ROWLAND

FIGURES OF FOO, which was put out by

Jack Hughes some time ago, is one of the best mathematical effects with apparatus that I know. I was told about the effect before I saw it performed or advertised, and I began to try to discover a method of working it at once. When, later on, I finally learned the original method I discovered that my own was rather different. Quite naturally I prefer my own, for although the performer is required to do a little calculation before he knows the final answer, it will still fool anyone who knows the original, because none of the chosen figures appears in any definite order in the answer. As the original effect has been on sale for some time, I do not think that any harm can be done if my method is published now.

The effect, of course, is that a spectator is given several flat strips of wood with a different column of figures printed or painted on both sides of each. He is asked to arrange them in any order, by fitting them into a corresponding number of slots in a simple stand, so that an ordinary addition sum consisting of several large numbers is formed. (I prefer to use cardboard strips which can be easily clipped across a small blackboard, and I use four strips with six figures on each side—though the number of strips and the number of figures on each can depend on the individual performer.) The spectator is finally asked to add the numbers together. Although he might be able to do this very quickly, you are always able to announce the total before he has finished, and apparently without giving more than a casual glance at the arrangement of the figures.

As the left-hand column can be chosen in eight different ways, the next one in six, the next in four, and the last in two, there are actually 8 x 6 x 4 x 2, or 384, different ways of arranging them all, and a corresponding number of possible totals.

The figures on both sides of the strips which I use are shown below, and this illustration can also be taken to represent two possible arrangements. The two different answers are therefore shown underneath as well.

4- |
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7 |
5" |
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(o |
I |
9 |
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8 |
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Z |
7 |
Z |
A- |
3 |
7 |
3 | |||||

5" |
3 |
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7 |
7 |
S |
(o | |||||

/ |
O |
8 |
9 |
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4- | |||||

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7 |
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3 Z |
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The figures in each column appear to have been chosen quite casually but as a matter of fact you will find that the total of any separate column is exactly six times the fourth figure from the top. It follows from this (and it might be a little sur prising to some of you) that the total of any possible arrangement of all the strips is always six times the whole number on the fourth row. Thus, the total of the first of these particular arrangements which I have illustrated is six times 5367, and the total of the second is six times 7562.

All that you have to be able to do then is to glimpse the number on the fourth row and multiply it quickly by six. This can be done under cover of writing the answer on a separate blackboard, but it is an added advantage to be able to do it mentally. You will find that this is not too difficult, for you have only a four-figure number to remember.

It is not necessary, of course, for the figures which give you the clue to the total to be on the fourth row; they could just as well be on any other. They are not very likely to be detected by anyone because the only column of figures which produces a multiple of six in the ordinary process of addition is the one on the extreme right. The other separate totals are increased each time by the addition of the tens figure which is carried forward from the right.

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