Find the Bogus Coin

May 13, 2000

RAY: Imagine that you have in front of you fifty coins. They all look exactly alike except one of them is a fake. Because it's a fake, it weighs a couple of grams more than a real coin. So, if you had a balance scale, and you knew which was the bogus coin, you would put it on one side of the scale, a good coin on the other side...

TOM: ...and it would be immediately obvious from this imbalance which was the phony coin, because it's heavier than a real coin.

RAY: Right. Knowing that, you have in front of you fifty coins -- one of which is bogus. The question is, what is the fewest number of weighings on a balance scale,that you need to perform, to determine which coin is bogus?

TOM: And, Part B of the puzzler: How can you do it in four?


RAY: Now, at first blush, you would think, because of other puzzlers of this ilk, that you would divide the 50 coins in half, and 50 is conveniently divided in half, right?

TOM: Yeah. So, you'd do 25 and 25. That's weighing number one.

RAY: Right.

TOM: You find out that it's on the left side.

RAY: Then you do 12 and 12 with one left over.

TOM: And --

RAY: Assume the worst case scenario.

TOM: Worst case scenario, one of them's heavier.

RAY: Right.

TOM: That's two.

RAY: Right.

TOM: Six and six.

RAY: Right.

TOM: That's three. Three and three. That's four.

RAY: And then --

TOM: And you're done for. It takes five.

RAY: Right.

TOM: No matter how you do it using that system, it don't work.

RAY: So, you had to come up with something a little more clever.

TOM: Yeah.

RAY: And what you do is divide the coins into three piles.

TOM: Oh, my God!

RAY: Two piles of 17 and one of 16.

TOM: Yeah.

RAY: And so, you take the two piles of 17 and you put those on the scale, and you keep the 16 pile aside, right?

TOM: Yeah.

RAY: Right away, you can see that you're going to eliminate not half the coins, but two thirds of the coins.

TOM: Oh, man. It's so beautiful, isn't it? It's beautiful.

RAY: So, let's assume that one of the 17 is the heavier one. You throw everything else away.

TOM: That's right.

RAY: And now --

TOM: And you've only made one weighing.

RAY: You've only made one weighing.

TOM: And you've narrowed it down to 17.

RAY: You've narrowed it down to 17 coins, OK? Now, you could divide the 17 in half, but better still, divide it thirds --

TOM: Why not keep doing what you're doing?

RAY: -- again.

TOM: This is beautiful. It took me a while to figure this out.

RAY: Yeah, well, with all the hints I gave you.

TOM: So, you divide it in thirds.

RAY: So, you've got six and six and five.

TOM: Yeah.


TOM: And that's the second weighing. No, you haven't done anything yet.

RAY: You haven't done anything, but you're going to put the six and six on.

TOM: Right.

RAY: OK? And you can see very clearly --

TOM: That you got it.

RAY: That you're going to be able to do this, because --

TOM: Three and three.

RAY: Three and three.

TOM: And one and one, and that's it.

RAY: And then, and bingo! And the key is, once you figure out the idea that you're going to divide it into three piles and not two, it jumps right out at you.

TOM: I always thought that the binary search was the only way to go.

RAY: Well, there you go. Who's our winner this week?

TOM: The winner is Judy Shalito. I like that name. From Mt. Pleasant, South Carolina.

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