TO ASCERTAIN A NUMBER THOUGHT OF
Every schoolboy knows the old puzzle: Think of a number; double it; add 10, divide by 2, subtract number thought of; and 5 left. Here is a great improvement upon that problem, which I have seen puzzle some excellent accountants.
Think of a number; multiply by 3; if the result is odd, add 1 and divide by 2; multiply by 3; if result be odd, add 1, and again divide by 2. By how many 9’s is the result divisible?
On receipt of that information you at once give the number thought of. One of the most puzzling features of the trick is that no 9’s are obtainable in the result should either 1, 2, or 3 be thought of, as the following will show:—
| Number thought of | 1 | 2 | 3 |
| multiply by | 3 | 3 | 3 |
| 3 | 9 | ||
| Add | 1 | 1 | |
| Divide by 2 | 4 | 6 | 10 |
| 2 | 3 | 5 | |
| Multiply by | 3 | 3 | 3 |
| 9 | 15 | ||
| Add | 1 | 1 | |
| Divide by 2 | 6 | 10 | 16 |
| 3 | 5 | 8 |
As will be seen, none of these results is divisible by 9, yet the number thought of is correctly given in each instance.
Solution.—When the number thought of is multiplied by 3, you ask the question, “Is the result odd or even?” If the answer is “odd,” make a mental note of one; then proceed. “Add one and divide by two. Is the result odd or even?” If the answer is again “odd,” make a mental note of two; and proceed. “Add one and divide by two. How many nines are obtainable in the result? I do not want to know what the surplus is.”
The above figures illustrate that when 1 is the number thought of there is only an addition of 1. When 2 is the figure, no addition is required to the first result; but the second result being 9, 1 is added and two noted, which, of course, is the figure thought of. When 3 is thought of two additions are necessary, one to the 9 and one to the 15, making a total of three to be remembered, which represents the original number. When 4 or any succeeding number is thought of the final result is always divisible by 9, and in your mental calculation each 9 must represent 4, to which you add the figures you have previously noted.
Examples.
Number thought of 4 × 3 = 12 ÷ 2 = 6 × 3 = 18 ÷ 2 = 9.
Here we have one 9, which represents 4, the number thought of.
Number thought of 7 × 3 = 21 + 1 = 22 ÷ 2 = 11 × 3 = 33 + 1 = 34 ÷ 2 = 17. From which is obtainable only one 9, which represents 4, to which you add 1 for the first addition 51 of 1, and 2 for the second addition, making a total of 7, the number thought of.
Number thought of,
| 11 | ||
| × | 3 | |
| 33 | ||
| + | 1 | note 1 |
| ÷ 2 | 34 | |
| 17 | ||
| × | 3 | |
| 51 | ||
| + | 1 | note 2 |
| ÷ 2 | 52 | |
| 26 | two 9’s = 8 = 11 |