Thursday, June 12, 2014

40 Pounds in the Balance

Problem

You are given a mystery ball and are told that it has a whole-number weight between 1 and 40 lbs (inclusive).
You are then given a balance scale are are allowed to pick four weights, each weighing any whole-number amount. You much choose these weights so that no matter what the mystery ball weighs, you will be able to determine its weight using just the balance scale and these four weights.
What weights should you choose?

Solution

The weights should be 1, 3, 9, and 27 lbs.

With the 1 lb weight alone, You can only determine a mystery ball of weight 1 lb.

With the 1 and 3 lb weights alone, you can determine a mystery ball of between 1 and 4 lbs. For example, if the mystery ball weighed 4 lbs, you could determine this by putting the 1 and 3 lb weights on one side of the balance, and the mystery ball on the other side. As another example, if the mystery ball weighed 2 lbs, you could determine this by putting the 3 lb weight on one side of the balance, and the mystery ball plus the 1 lb weight on the other side.

With the 9 lb weight in the mix, we can now determine the weights of a mystery ball between 1 and 13 lbs. And finally, with the 27 lb weight, we can determine any weight between 1 and 40 lbs.

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