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. How Many Counters Have I in My Hands? A person having an equal number of counters in each hand, it is required to find how many he has altogether. Suppose he has 16 counters, or 8 in each hand. Desire him to transfer from one hand to the other a certain number of them, and to tell you the number so transferred. Suppose it be 4, the hands now contain 4 and 12. Ask him how many times the smaller number is contained in the larger; in this case it is three times. You must then multiply the number transferred, 4, by the 3, making 12, and add the 4, making 16; then divide 16 by the 3 minus 1; this will bring 8, the number in each hand. In most cases fractions will occur in the process; when 10 counters are in each hand and if four be transferred, the hands will contain 6 and 14. He will divide 14 by 6 and inform you that the quotient is 2-1/3. You multiply 4 by 2-1/3, which is 9-1/3. Add four to this, making 13-1/3 equal to 40/3. Subtract 1 from 2-1/3, leaving 1-1/3 or 4/3. Divide 40/3 by 4/3, giving 10, the number in each hand. The Three Travelers.