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

against three. Again whichever of the three tips the scale is where the heavier ball is. So of the three balls (where one of them is heavier than the other two) weigh one against one. If they balance, the heavier ball is the remaining one. If they don’t balance, whichever ball tips the scale is the heavier one.
    Method 2: First weigh four balls with four balls. Suppose they balance. Then weigh the remaining four balls—two balls against two balls. Whichever two balls tip the scale downward include the heavier ball. So weigh one of those balls against the other. Whichever ball tips the scale downward is the heavier ball.
    Suppose when we weigh the four balls against four balls, they don’t balance. The four balls that tip the scale downward have one of the balls that is heavier. So take these four balls and weigh two against two. One of the two balls will tip the scale downward. Whichever of the two balls tips the scale downward contains the heavy ball. Now weigh one of those balls with the other. The one that tips the scale downward is the heavy ball.
    Method 3: First weigh three against three. If they balance, then weigh the other three against three. Certainly one side will be heavier so weigh the three balls where one is heavier, one against one. If they balance, it’s the remaining ball that is heavier. If they don’t balance, the ball that tips the scale downward is heavier.
    If the original three against three don’t balance, take the three balls that tip the scale downward and weigh two of those balls, one on one side, the other on the other. If they balance, it’s the remaining ball. If they don’t balance, it’s the ball that tips the scale downward.
    147. The man is 52 and his wife is 39.
    Denote the man’s age now as M , the wife’s age now as W , the man’s age when he was as old as the wife now as m, the wife’s age when the man was as old as she is now as w .
    Then we get
    (1) M + W = 91
    (2) m = W (since the man was then as old as the wife now)
    (3) M = 2 w (since the man is twice as old as the wife was)
    The key thing to realize is that the difference in ages between the man and his wife now is the same difference then or at any other time. That is,
    (4) M - W = m - w
    So substituting (2) and (3) into (4), we get
    M - W = W - M /2 or M - W = (2 W - M )/2, which gives us 2 M - 2 W = 2 W - M , so we get
    (5) 3 M = 4 W or M = 4 W /3
    We substitute (5) into (1) and we get:
    4 W /3 + W = 91; 7 W /3 = 91 and W = 273/7 = 39. From (1) we get M + 39 = 91, and so M = 52.
    148. 45/50 or 90 percent
    n = nickels, p = pennies, q = quarters, d = dimes
    We have 5 n + p + 10 d + 25 q = 100 (since the total is 100 cents) and n + p + d + q = 50 (since the total is 50 coins).
    Subtract the two equations:
    We get 4 n + 9 d + 24 q = 50.
    Suppose q = 1. Then 4 n + 9 d + 24 = 50, and so 4 n + 9 d = 26.
    The only way this is possible is if d = 2 and n = 2.
    Suppose q = 2. Then 4 n + 9 d + 48 = 50, and we get 4 n + 9 d = 2, which is impossible.
    So we have 1 quarter, 2 dimes, and 2 nickels, which leaves 45 pennies since the total number of coins is 50. If I drop 1 penny, we have the probability as 45/50 or 90 percent.
    149. (e) 108
    Translate: The number of apples that Bill bought = B, that Harry bought = H, and that Martin bought = M . “Bill bought four times as many apples as Harry” translates to B = 4 H. Similarly B = 3 M. “ Bill, Harry, and Martin purchased a total of less than 190 apples” translates to B + H + M < 190. You will find that manipulating these equations, we get B < 120. However, because H and M are integers, B = 108 and not 119!
    Here’s the complete solution:
    (1) B + H + M < 190
    (2) B = 4 H
    (3) B = 3 M
    Substituting (3) into (1) we get
    (4) 3 M + H + M < 190
    From (2) and (3), we get
    (5) 3 M = 4 H and so
    (6) H = 3 M /4
    Substituting (6) into (4) we get
    (7) 3 M + 3 M /4 + M < 190
    This becomes
    (8) 19 M /4 < 190 and thus
    (9) M /4 < 10, so M < 40. So at this point you might think that M