The (almost really) Complete Works of Lewis Carroll

November 19, 1866

To the Editor of the Pall Mall Gazette

Sir,—The magical system of betting, the secret of which Messrs. H. and J. Smith offer to the world on such reasonable terms, has probably been known, and practised, ever since betting has been in existence. It is applicable to almost every event on which bets are made, and it may be mathematically demonstrated that, provided all the bets are paid, winning is a certainty. I chanced upon the principle myself some years ago, and, in the hope that it may serve to deter some from throwing away their money, I now beg to offer it to your readers gratis.

The rule may be stated thus:—“Write all possible events in a column, placing opposite to each the odds offered against it: this will give two columns of figures. For the third column add together the odds in each case, and find the least common multiple of all the numbers in this column. For the fourth column divide this least common multiple by the several numbers in the third column. For the fifth and sixth columns multiply the original odds by the several numbers in the forth column. These odds are to be given, or taken, according as the sum total of the sixth column is greater or less than the least common multiple.” The last two columns give the relative amounts to be invested in each bet.

An example will make this clear. Suppose that in a race about to be run there are four horses in the betting, the odds being 3 to 2 on the favourite, which is equivalent to 2 to 3 against. The least common multiple of the third column is 60, and the sum total of the last 68, and as this is greater than 60, the odds in this case are all to be given in the relative amounts given in the fith and sixth columns. Suppose, for example, that I multiply these columns by 10, and make the bets in pounds; that is, I take £360 to £240 on A., I give £480 to £120 against B., and so on. Now suppose C. to win the race; in this case I lose £500, and win £(360 + 120 + 60 + 40) = £580. It will be found on trial that I win the same sum, £80, in each of the five events.

If all betting men tried to work this system, they would either be all offering odds or all taking odds on each event, and so no bets could be made. But the fact that this system of winning is ever possible arises from the odds being unevenly adjusted, so that they do not represent the real chances of the several events. Supposing this system to be applied only in cases where the odds were evenly adjusted, the sum total of the sixth column would always be equal to the least common multiple, and thus, whether the odds were given or taken, the concluding entry in every betting-book would be “Gain=Loss=Nil”—a most desirable result,—I am, Sir, your obedient servant,

November 20, 1866

To the Editor of the Pall Mall Gazette

Sir,—In the arithmetical example of the rule which I sent you on this subject, 30 should have been given instead of 60 as the least common multiple. The mistake does not affect the validity of the rule, as any common multiple will serve the purpose.—I am your obedient servant,

November 21, 1866

To the Editor of the Times

Sir,—As you have thought my communication to the Pall-mall Gazette on the above subject worth republishing in your columns, will you allow me to correct a mistake in the arithmetical example? It should stand thus:—

The least common multiple of the third column is 30, not 60. The truth of the rule is not affected by this, as any common multiple would serve the purpose. I am, Sir, your obedient servant,