##### Oct 06, 2003

**RAY**: Remember the puzzler from last week? You have eight coins, and one is heavier than the other seven. How can you find it on a balance scale in two weighings?

Well, hopefully you learned something from last week's puzzler that will enable you to solve this week's puzzler.

Here it is.

You have 50 coins. One of them is bogus, and heavier than the other 49. You have the same balance scale that you used to solve last week's puzzler.

You can figure out which coin is bogus in four weighings. The question is, how?

**RAY**: Here's how:

**Step 1**: Divide the coins into three piles. A pile of 17, another pile of 17 and a pile of 16.

**Step 2**: Weigh the two piles of 17 on your balance beam. This is your first weighing.

For the moment, let's assume that one pile is heavier. We know that the bogus coin is in that group. In that case, we need to take the following steps:

Divide the heavier pile of 17 into a pile of 6, another pile of 6 and a pile of 5.

Weigh the two piles of 6. Let's assume that one pile is heavier. In this case, take the heavier pile of six and break it into three piles of two. For your third weighing, weigh two of the piles. If one pile is heavier, weigh those two coins against each other to find the bogus coin. If not, weigh the pile of two that you put aside to find the heavier coin.

Let's go back a step and see what happens if those two piles of six you weighed were equal. In this case, you know that the heavy coin is in the group of 5 coins. So, take the pile of 5 and break it into a pile of 2, a pile of 2 and 1 coin. Weigh the 2 piles of 2 against each other. If they're equal, you know the bogus coin is the one you didn't weigh. If they're not equal, take the heavier pile of 2, split it and weigh those 2 coins to find the bogus one.

Now, let's go all the way back to Step 2. Let's now assume that two piles of 17 coins balanced when you weighed them and the heavy coin was in the group of 16. In that case, you'd take that pile of 16 and break it into a pile of 5, another pile of 5 and a pile of 6, and your next step would be to weigh the 2 piles of 5 against each other. There are two possible outcomes here:

If the 2 piles of 5 are equal, you know the heavy coin is in the pile of 6. In this case, break the 6 coins in to 3 piles of 2. Weigh 2 piles against each other. If they're equal, you know the bogus coin is in the pile you didn't weigh, and you can find the coin in one more weighing. If they're not equal, take the heavier pile of 2, split it and weigh those 2 coins to find the bogus one.

If the 2 piles of 5 are unequal, you know the bogus coin is in the heavier pile. In that case, take the heavier pile and divide it into a pile of 2, another pile of 2 and a pile of just 1. If you can't figure out the bogus coin from here in no more than two weighs, you're in big trouble!

Pretty neat, huh?