Odd Coin Problems

Jack Wert

Below is a much shorter description of the Odd Weight Coin problem:

It is assumed you know that you can identify an odd weight coin, knowing that it is heavier (or, that it is lighter) than the others according to the 3ⁿ equation, "n" being the number of weighings to identify the odd coin in 3ⁿ coins. For example, it takes only 2 weighings to find the odd coin in 9 coins, or 3 weighings for 27 coins, etc. However, finding the odd coin in a group, when it is not known whether it is heavier or lighter, but just a different weight - is a bit more difficult.

This is a general solution that involves no labeling of coins, and can be handled mentally for a group of coins even as high as 1,092, in seven weighings. It can be applied to the classic *12 coin problem on up.

Given (3ⁿ-3)/2 coins, divide them into three equal main groups, and then each main group into (n-1) sub-groups of 3^n-2, 3^n-3, ..., 3⁰ coins. Place a main group on each pan, and one on the table. Observe the condition of the balance. This is the first weighing.

Rotate the largest sub-groups, moving the left pan sub-group onto the table: the right pan sub-group onto the left pan; the previous table sub-group onto the right pan. Observe the new condition of the balance. This is the second weighing.

If the condition of the balance changes, you will know which sub-group contains the odd coin, and its relative weight. In this case, clear the balance of the other coins, and continue weighings according to the "3ⁿ" coin solution. If the condition of the balance remains the same, discard the sub-groups just rotated.

Continue this rotation process with the largest remaining sub-groups until you get a change in the condition of the balance, or until you have only one coin at each position, then rotate them and you will have the odd coin and its relative weight. Problem solved!

Solution to the 12 coin problem

Divide the coins into three 4 coin main groups, and then each of these into sub-groups of 3 and 1 coins. Place two of the main groups on the pans of the balance. Observe the condition of the balance. This is the first weighing.

Rotate the 3 coin groups, and observe the condition of the balance. This is the second weighing.

If there is no change, all 3 coin groups are good coins. Clear them from the balance, and rotate the single coins. This is the third weighing, and will identify the odd coin and determine its relative weight. Problem solved!

If there is a change, it will identify the group that contains the odd coin, and determine its relative weight. Clear the balance of all other coins, and put one coin from the odd 3 coin group on each pan, and the third one on the table. This is the third weighing, and will identify the odd coin. As you already know its relative weight - Problem solved!

1. B. Bundy, The Oddball Problem
2. Weighing 12 coins, Dyson and Lyness' solution
3. W. McWorter, Weighing 12 coins
4. D. Newman, THOUGHT LESS MATHEMATICS
5. Weighing with counterbalances
6. J. Wert, Odd Coin Problems
   • Odd Coin Problems, a shortened exposition
7. Six Balls, Two Weighings

Copyright © 1996-2008 Alexander Bogomolny