The Magician's Own Book, or, the Whole Art of Conjuring / Being a complete hand-book of parlor magic, and containing over one thousand optical, chemical, mechanical, magnetical, and magical experiments, amusing transmutations, astonishing sleights and subtleties, celebrated card deceptions, ingenious tricks with numbers, curious and entertaining puzzles, together with all the most noted tricks of modern performers.

by George Arnold; Frank Cahill

THE VISITORS TO THE CRYSTAL PALACE.

In a family consisting of 8 young people, it was agreed that 3 at a time should visit the Crystal Palace, and that the visit should be repeated each day as long as a different trio could be selected. In how many days were the possible combinations of 3 out of 8 completed?

We must multiply 8 × 7 × 6, and also 3 × 2 × 1, and divide the product of the former, 336, by the product of the latter, 6; the result is 56, the number of visits, a different three going each time. So much gratified were they with the results of their agreement, that they wished to be allowed another series of visits, to be continued as many days as they could group 3 together in different order when starting. If Paterfamilias had granted such permission he would have had to wait 56 multiplied by 3 × 2 × 1, or 336 days, before this "new series" of visits would have come to a finis.

HOW MANY CHANGES CAN BE GIVEN TO 7 NOTES OF A PIANO?

That is to say, in how many ways can 7 keys be struck in succession, so that there shall be some difference in the order of the notes each time?

The result of multiplying

7 × 6 × 5 × 4 × 3 × 2 × 1

is 5,040, the number of changes.

THE ARITHMETICAL TRIANGLE.