Maths Puzzles

You have 12 coins, 11 of which are the same, but one of which is a fake. The fake can be detected because it has a different weight to the other 11, and you have a set of scales to help you work out which it is. But the scales will tell you only which side is heavier or lighter, you don't know whether the fake is too heavy or too light, and you have only three uses of the scales to come to your conclusion. Can you say which is the fake coin, and whether it is too heavy or too light?

Hint: consider not just where the fake coin can be, but where it can be depending on if it is heavier or lighter than the other coins.



Solution:

I have denoted the 12 coins by the letters of the alphabet, ABCDEFGHIJKL. At each weighing there can be three outcomes: the left side is heavier than the right; the scales are balanced; the right side is heavier than the left. Whatever the outcome is, the possible set of solutions is narrowed each time, and I have included this set at each stage of the solution. For example {A+, B+, F−} means that the fake coin is either A (too heavy), B (too heavy) or F (too light).

Every possible solution to the problem can be reached by a unique path. There are several equivalent ways of doing this, but the basic method is the same each time. One possible route map is shown below...

Weigh ABCD vs. EFGH

Puzzles