## Tuesday, September 23, 2008

### A Farmer and His Stones

A farmer had a stone that he used to measure grain on his scale. One day his neighbor borrowed the stone, and when he returned, it was broken into four pieces. The neighbor was very apologetic, but the farmer thanked the neighbor for doing him a big favor. The farmer said that now he can measure his grain in one pound increments starting at one pound all the way to forty pounds (1, 2, 3, 17, 29, 37, etc.) using these four stones.

How much do the four stones weigh?

Stuck? There are some hints below:

Hint 1: The stones have broken (miraculously) into four parts that are each X pounds each, where X is a whole number. In other words, there's no stone that weighs 5 and 1/2 pounds, but there could be one that weighs 5 pounds.

Hint 2: Each stone is different in size.

1. Well, they have to add up to 40, and you have to be able to build all the numbers from 1 to 40 with combinations of the 4 (adding -- putting more stones opposite the grain -- and subtracting -- adding the stones to the grain side of the scale). You're going to need various degrees of resolution, so it seems like you'll need a 1-pounder, then increasing weights.

If you guess that the first one is 1, then it seems reasonable that the second could be 3, which is the highest number you could use to make 2 (3-1). You can then make 3 and 4 easily. To make 5, you the highest number you can use is a 9 (9-3-1). You can make 6-13 with these. The highest number you can use to make 14 is 27 (27-9-3-1). You can make numbers 14-40 with these four stones: 1, 3, 9, and 27 pounds.

I can't put my finger on the significance of the fact that the stone weights are in powers of 3. Any thoughts?

2. As usual for these types of problems, there's a mathematical principle that guides the question.

But I have no idea what it is! ;-)

But you're right, Abe. The four stones are 1, 3, 9 and 27.

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