PUZZLE OF THE 12 COINS


THE PROBLEM

The problem is to find a way to determine if there is a fake coin in a batch of 12 coins. The biggest problem is, you do not know whether the fake coin is lighter or heavier than the genuine ones. You are given a scale to weigh the coins.

The challenge is to determine the fake coin, if any, by weighing the coins only 3 times AND to determine whether the fake coin is heavier or lighter than the genuine ones.


THE SOLUTION

To help you understand easier, status of the various coins are given at all stages of the answer. Black means confirmed genuine, blue means unconfirmed and red means the fake coin.

Weigh # 1

Weigh coins # 1, 2, 3 & 4 against coins # 5, 6, 7 & 8. (1,2,3,4,5,6,7,8,9,10,11,12)

    If they balance, then all the eight coins are genuine ones. Proceed to Weigh # 2A. (1,2,3,4,5,6,7,8,9,10,11,12)

    If they do not balance, then one of these eight coins is a fake and coins # 9 to 12 are genuine. Proceed to Weigh # 2B. (1,2,3,4,5,6,7,8,9,10,11,12)

Weigh # 2A

Weigh any 3 coins (# 1 to 8) against coins # 9, 10 & 11. (1,2,3,4,5,6,7,8,9,10,11,12)

    If they balance, then coins # 1 to 11 are genuine ones. Proceed to Weigh # 3A. (1,2,3,4,5,6,7,8,9,10,11,12)

    If they do not balance, then coins # 9 to 11 are suspect. Note whether these coins are heavier or lighter than the genuine ones. Go to Weigh # 3B. (1,2,3,4,5,6,7,8,9,10,11,12)

Weigh # 2B

Assume coins # 1 to 4 is heavier than coins # 5 to 8. Weigh coins # 1, 2 & 5 against coins # 3, 4 & 6. (1,2,3,4,5,6,7,8,9,10,11,12)

    If they balance, coins # 1 to 6 are genuine and either coins # 7 or 8 is a fake. Proceed to Weigh # 3C. (1,2,3,4,5,6,7,8,9,10,11,12)

    If they do not balance and assuming coins # 1, 2 & 5 is heavier than coins # 3, 4 & 6, then coins # 3 to 5 are genuine. Proceed to Weigh # 3D. (1,2,3,4,5,6,7,8,9,10,11,12)

Weigh # 3A

Weigh any coins from #1 to 11 against # 12. (1,2,3,4,5,6,7,8,9,10,11,12)

    If it balance, all 12 coins are genuine. (1,2,3,4,5,6,7,8,9,10,11,12)

    If it does not balance, then coin # 12 is a fake and this weighing will also determine whether it is lighter or heavier than a genuine coin. (1,2,3,4,5,6,7,8,9,10,11,12)

Weigh # 3B

Weigh coin # 9 against # 10. (1,2,3,4,5,6,7,8,9,10,11,12)

    If it balance, coin # 11 coins is a fake and Weigh # 2A will determine whether it is lighter or heavier than a genuine coin. (1,2,3,4,5,6,7,8,9,10,11,12)

    If it does not balance, then either of these coins is a fake depending on the result of Weigh # 2A. (1,2,3,4,5,6,7,8,9,10,11,12)

Weigh # 3C

Weigh coin # 7 against coin # 8. (1,2,3,4,5,6,7,8,9,10,11,12)

    If it balance, check the weighing machine! (1,2,3,4,5,6,7,8,9,10,11,12)

    If it does not balance, then the lighter one is a fake. (1,2,3,4,5,6,7,8,9,10,11,12)

Weigh # 3D

Weigh coin # 1 against coin # 2. (1,2,3,4,5,6,7,8,9,10,11,12)

    If it balance, coin # 6 is a fake and it is lighter than a genuine coin. (1,2,3,4,5,6,7,8,9,10,11,12)

    If it does not balance, then the lighter one is a fake. (1,2,3,4,5,6,7,8,9,10,11,12)


THE ALTERNATE SOLUTION

I found another solution to this problem in a book on puzzles in my old school library. I have forgotten the bulk of it but I do remember the solution was a very elegant and poetic one. It tells the story of this challenge in a poem and it ended with the boy telling the mother to solve the challenge with the unforgettable words:

M A ! DO       L I K E

M E T O       F I N D

F A K E        C O I N!

It has been 30 years and these few words still sticks to my mind when all else is gone!