A Person having an even Number of Counters in one Hand, and an odd Number in the other, to tell in which hand each of them is.

Desire the person to multiply the number in his right hand by three, and the number in his left by two.

Bid him add the two products together, and tell you whether the sum be odd or even.

If it be even, the even number is in the right hand; but if it be odd, the even number is in the left hand.

Example I.

No. in right hand. No. in left hand. 18 7

3 2

64 54 14

68 sum of the products.

Example 11.

No in right hand No. in left hand. 7 18

3 2

91 21 36

57 sum of the products.

A Person having fixed on a Number in his Mind, to tell him what

Number it is.

Bid him quadruple the number thought on, or multiply it by 4; and having done this, desire him to add 6, 8, 10, or any even number you please, to the product; then let him take the half of this sum, and tell you how much it is; from which, if you take away half the number you desired him at first to add to it, there will remain the double of the number thought on. Example.

Suppose the number thought on is.......•.... 5

The quadruple of it is...................... 20

8 added to the product is...................28

And the half of this sum .................... 14

4 taken from this leaves ............... 10.

Therefore 5 was the number thought on.

Another Method of discovering a Number thought on.

After the person has fixed on a number, bid him double it, and add 4 to that sum; then let him multiply the whole by 5, and to that product add 12; desire him also to multiply this sura by 10, and after having deducted 302 from the product, to tell you the remainder, from which, if you cut off the last two figures, the number that remains will be the one thought on. Example.

Let the number thought on be ............ 7

Then the double of this is ................ 14

And 4 added to it makes .................. 18

This multiplied by 5 is................ 90

And 12 add d to it is .................... 102

And this multiplied by 10 is.............. 1020

From which deducting .................... 302

There remains .......................... 718,which, by striking off the last two figures, gives 7, - the number thought on.

To tell the Number a Person has fixed upon, without asking him any Questions.

The person having chosen any number in his mind, from I to 15, bid him add one to it, and triple the amount. Then,

If it be an even number, let him take the half of it, and triple that half; but if it be an odd number, he must add 1 to it, and then halve it, and triple that half.

In like manner let him take the half of this number, if it be even, or the half of the next greater, if it be odd; and triple that half.

Again, bid him take the half of this last number, if even, or of the next greater, if odd; and the half of that half in the same way; and by observing at what steps he is obliged to add 1 in the halving, the following table will show the number thought on:

1-0-0 -4-8

2-0-0 - -13- 5

3-0-0 - 3-11

1-2-0 - 2-10

1-3-0 -8-0

1-2-3 - 6-14

2-3-0 -1-9

0-0-0 -15- 7

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, etc.

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.