Thus, if he be obliged to add 1 only at the first step, or halving, either 4 or 8 was the number thought on; if there were a necessity to add 1 both at the first and second steps, either 2 or 10 was the number thought on, &c.

And which of the two numbers is the true one may always be known from the last step of the operation; for if 1 must be added before the last half can be taken, the number is in the second column, or otherwise in the first, as will appear from the following examples:

Suppose the number chosen to be 9
To which, if we add 1
The sum is 10
Then the triple of that number is 30
1. The half of which is 15
The triple of 15 is 45
2. And the half of that is 23
The triple of 23 is 69
3. The half of that is 35
And the half of that is 18

From which it appears, that it was necessary to add 1 both at the second and third steps, or halvings; and therefore, by the table, the number thought on is either 1 or 9. And as the last number was obliged to be augmented by 1 before the half could be taken, it follows also, by the above rule, that the number must be in the second column; and consequently it is 9.

Again, suppose the number thought on to be 6
To which, if we add 1
The sum is 7
Then the triple of that number is 21
1. The half of which is 11
The triple of 11 is 33
2. And the half of that is 17
The triple of 17 is 51
3. The half of that is 26
And the half of that half is 13

From which it appears, that it was necessary to add 1 at all the steps, or halvings, 1, 2, 3, therefore, by the table, the number thought on is either 6 or 14.

And as the last number required no augmentation before its half could be taken, it follows also, by the above rule, that the number must be in the first column; and consequently it is 6.

A curious Recreation, usually called—The Blind Abbess and her Nuns.

A blind abbess visiting her nuns, who were twenty-four in number, and equally distributed in eight cells, built at the four corners of a square, and in the middle of each side, finds an equal number in every row, containing three cells. At a second visit, she finds the same number of persons in each row as before, though the company was increased by the accession of four men. And coming a third time, she still finds the same number of persons in each row, though the four men were then gone, and had each of them carried away a nun.

Fig. 1. Fig. 2. Fig. 3.
333 252 414
3 3 5 5 1 1
333 252 414