The Puzzle:

I have twenty-seven pennies. Twenty-six of the pennies weigh exactly the same. The twenty-seventh penny, however, weights slightly more than the other twenty-six. I only have time to make three weighing’s using a balance scale of the type shown in the picture. How should I proceed so that I can find the odd-weighing penny?

The Solution:

I start by dividing the twenty-seven pennies into three groups of nine. I put one group of nine pennies on one side of the balance scale and a second group of nine pennies on the other side of the balance scale. If the two sides of the scale balance, the heavy penny in in the third group of three not on the scale. If the two sides of the scale do not balance, the heavy penny is in the heavier group of nine on the scale. In any case, I have narrowed the number of pennies to nine and I have used one of my three weighing’s.

I now take the group of nine pennies that I have identified as having the heavy penny into three groups of three and I repeat that procedure above; that is, I put one group of three pennies on one side of the scale and a second group of three pennies on the other side of the scale. If the two sides balance, the heavy penny is in the group of three not on the scale. If the two sides of the scale do not balance, I pick the heavier group of three pennies as containing the heavier penny. In any case, I have now narrowed the number of pennies to three and I have used two of my three weighing’s with one more to go.

Finally, I take one of the three pennies and put it on one side of the scale and another of the three pennies on the other side of the scale. If the two sides of the scale balance, the heavier penny is the one not on the scale. If the two sides of the scale do not balance, I pick the heavier penny from the heavy side of the scale. In either case, I now have the heavy penny and I have used my three allotted weighing’s.

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