Free World's 200 Hardest Brain Teasers by Dr. Gary R. Gruber Page A

= 39. And from (3), B becomes 117. But from (6) we wouldn’t have H as a whole number! The greatest number less than 40 which the whole number M could be for (6) to be true is M = 36. Then H = 27, and from (3), B = 108.
    150.

    Let’s see what numbers would be on the vertices of the triangle. Represent those numbers by a, b, and c. Let the numbers 1 through 9 be represented by a, b, c, d, e, f, g, h, i. Then we have
    (1) a + e + d + b = 17
    (2) b + f + g + c = 17
    (3) c + h + i + a = 17
    (4) a + b + c + d + e + f + g + h + i = 45 (since all the numbers 1 + 2 + 3…9 add up to 45)
    Adding (1), (2), (3) we get
    (5) 2 a + 2 b + 2 c + d + e + f + g + h + i = 51.
    Subtracting (4) from (5) we get a + b + c = 6. The only way this can be possible is if the numbers for a , b , and c are 1, 2, 3.
    So start with the numbers 1, 2, and 3 at the vertex of the triangle. For the left side, since the numbers must add up to 17, the other two numbers on the left side must add up to 14. The only possibilities are 6 and 8 or 5 and 9. If it is 5 and 9, then the remaining numbers are 4, 6, 7, 8. For the bottom side, two of these numbers (of 4, 6, 7, 8) must add up to 17 - 5 = 12. The only way is 4 and 8. So then we’d have 6 and 7 left. For the right side, 1 + 3 + 6 + 7 adds up to 16. Similarly you can see that for the left side 6 and 8 also works and we get the triangle on the right.
    151. There are at least four different solutions.
    Think outside of the box. Draw lines outside of the square that contains the circles.
    Solution 1

    Solution 2

    Note that the two ends can be extended to form a continuous path. There are four points external to the 4 x 4 array.
    Solution 3

    Solution 4

    152. (c) There is the same amount of water in the alcohol as alcohol in the water.
    Let’s say the cup of alcohol you pour in the water contains c gallons (c could be 1/10, for example). Now the water bucket has a mixture of alcohol and water. Now you pour one cup (again, c gallons) of that mixture back into the alcohol bucket. Let’s say there are a gallons of alcohol and w gallons of water in that cup. So (1) a + w = c .
    Now the amount of water poured into the alcohol is w. The amount of alcohol in the water bucket is c - a, since we poured a gallons of alcohol into the alcohol bucket from the water bucket. From (1) we get c - a = w, so (c) is true.
    153. (e) octagon
    Here’s the figure:

    154. (d) e
    The sequence is arranged in pairs: st no jk gh
    Note that we have a pattern: gh (i) jk (lm) no (pqr) st
    The letters in parentheses increase by one letter in the sequence. Thus, the next two letters in the original sequence must be e, f, since the pattern will be preserved:
    ef (no letters) gh (i) jk (lm) no (pqr) st
    155.  x /2 + y
    You may have gotten the answer x /2 - y, but that’s how many people are left on the bus!
    x - (x/2 - y), which is x/2 + y, is the answer.
    156. LIGHTS, CAMERA, ACTION
    157. (b) 2
    Write an equation, where n is the number of nickels, d is the number of dimes, and q is the number of quarters: 5 n + 10 d + 25 q = 100. Start with the smallest numbers for n , d , and q that satisfy the equation. Start with n = 0, d = 0, then q = 4, which is not true since the most q can be is 3. Then try n = 0, d = 1. Then we get 25 q = 90, which doesn’t give us a whole number for q. You will find that if n = 1, d = 2, and q = 3, and if n = 3, d = 1, and q = 3, you will satisfy the equation. So there are two ways to make change.
    158. There was no extra dollar.
    Watch that your mind doesn’t deceive you. They thought the bill came to $30, but that doesn’t make it the amount that we want to work with.
    159. 95,999
    You may have thought the number was 96,666. But then you didn’t account for the 9 in the original number. So the least whole number is 95,999.
    160. facetious
    161. Four states
    Hawaii (Honolulu), Oklahoma (Oklahoma City), Indiana (Indianapolis), Delaware (Dover).
    162. (d) song : hymn
    Put the analogy in a very specific