A FOURTH WAY.
Tell the person to multiply the number thought of by itself; then desire him to add 1 to the number thought of, and to multiply it also by itself; in the last place, ask him to tell the difference of these two products, which will certainly be an odd number, and the least half of it will be the number required.
Let the number thought of, for example, be 10; which, multiplied by itself, gives 100; in the next place, 10 increased by 1 is 11, which, multiplied by itself, makes 121; and the difference of these two squares is 21, the least half of which, being 10, is the number thought of.
This operation might be varied by desiring the person to multiply the second number by itself, after it has been diminished by 1. In this case, the number thought of will be equal to the greater half of the difference of the two squares.
Thus, in the preceding example, the square of the number thought of is 100, and that of the same number, less 1, is 81; the difference of these is 19, the greater half of which, or 10, is the number thought of.