To Discover Two or More Numbers that a Person has Thought of.
FIRST CASE.
Where each of the numbers is less than 10. Suppose the numbers thought of were 2, 3, 5.
| EXAMPLE. | |
| 1. Desire him to double the first number, making | 4 |
| 2. To add one to it | 5 |
| 3. To multiply by 5 | 25 |
| 4. To add the second number | 28 |
| There being a third number, repeat the process. | |
| 5. To double it | 56 |
| 6. To add 1 to it | 57 |
| 7. To multiply by 5 | 285 |
| 8. To add the third number | 290 |
And to proceed in the same manner for as many numbers as were thought of. Let him tell you the last sum produced (in this case, 290). Then, if there were two numbers thought of, you must subtract 5; if three, 55; if four, 555. You must here subtract 55; leaving a remainder of 235, which are the numbers thought of, 2, 3, and 5.
SECOND CASE.
Where one or more of the numbers are 10, or more than 10, and where there is an odd number of numbers thought of.
Suppose he fixes upon five numbers, viz., 4, 6, 9, 15, 16.
He must add together the numbers as follows, and tell you the various sums:
| 1. The sum of the 1st and 2d | 10 |
| 2. The sum of the 2d and 3d | 15 |
| 3. The sum of the 3d and 4th | 24 |
| 4. The sum of the 4th and 5th | 31 |
| 5. The sum of the 1st and last | 20 |
You must then add together the 1st, 3d, and 5th sums, viz., 10 + 24 + 20 = 54, and the 2d and 4th, 15 + 31 = 46; take one from the other, leaving 8. The half of this is the first number, 4; if you take this from the sum of the 1st and 2d you will have the 2d number, 6; this taken from the sum of the 2d and 3d will give you the 3d, 9; and so on for the other numbers.
THIRD CASE.
Where one or more of the numbers are 10, or more than 10, and where an even number of numbers has been thought of.
Suppose he fixes on six numbers, viz: 2, 6, 7, 15, 16, 18. He must add together the numbers as follows, and tell you the sum in each case:
| 1. The sum of the 1st and 2d | 8 |
| 2. The sum of the 2d and 3d | 13 |
| 3. The sum of the 3d and 4th | 22 |
| 4. The sum of the 4th and 5th | 31 |
| 5. The sum of the 5th and 6th | 34 |
| 6. The sum of the 2d and last | 24 |
You must then add together the 2d, 4th, and 6th sums, 13 + 31 + 24 = 68, and the 3d and 5th sums, 22 + 34 = 56. Subtract one from the other, leaving 12; the 2d number will be 6, the half of this; take the 2d from the sum of the 1st and 2d, and you will get the 1st; take the 2d from the sum of the 2d and 3d, and you will have the 3d, and so on.