A "Magic" Trick

There is a so-called "magic trick" that goes like this:

"A farmer has five chickens. He foolishly buys two foxes." Pennies stand for the foxes and chickens. You have one penny in each hand, for each fox, and 5 pennies on the table, standing for each chicken. You close your hands into fists, so people never again see what is inside your hand until the end of the story.

"The farmer leaves for the night, telling the foxes not to take the chickens. The first thing the foxes do when the farmer leaves is take the chickens." You pick up the chickens one-by-one, alternating hands, symbolizing that the foxes are taking the chickens and splitting them evenly.

"Then the farmer returns. The foxes hear him coming, so they return the chickens." You put the five chickens/pennies back, one by one, alternating hands.

"But as soon as the farmer leaves, the foxes again take the chickens." As before, you pick up the pennies/chickens one by one alternating hands.

"But the farmer was suspicious of the foxes. This time he crept back without the foxes hearing him. And what did he find?" You open your hands. "The two foxes were sleeping together, and the five chickens were fine." There are two pennies/foxes in one hand and five pennies/chickens in the other.

How it is Done

I really shouldn't tell you how to do the "trick". After all, if we are expecting the students to figure out the trick, then you should be expected to do the same. There is no magic, no slight-of-hand, and nothing going on other than what was described above.

The puzzle of course is that, seeing how the chickens were taken and returned, you would expect each hand to have the same number of pennies (approximately). How did it get so off from even (5 to 2)?

What Does This Teach?

There's a concept or two, here, though I am not exactly sure what it is. The puzzle is completely mathematical. But there is no well-known concept here that I can find. It reminds though of a constant nagging problem that occurs in a variety of different disguises.

For example, determining the median of a set of numbers is straightforward when there are an odd number of numbers, but all of a sudden becomes problematic when there are an even number of numbers.

Or, the hundredth number in this set <1,3,5....> is around 200, but is it 199 or 201?

So maybe the concept is that when things should divide evenly but don't, you can't just ignore the error.

What made me think of this was a person trying to divide a pile of cards into three equal piles, correctly (and insightfully) dealing the cards into three piles, then getting the answer wrong because she went from right to left to right to left to right etc. and gave too many cards to the center pile.

How To

I first did the trick, then did it again. When someone said they knew how to do it, I let them try it. Mostly, people think they can do it, but they end up with 4 and 3, not 5 and 2. In both classes, I wandered off when someone else started doing the trick. I brought in a bag of pennies and started handing them out so that people could try it themselves. Actually, it was a student's idea, and I just happened to have a bag of pennies. But it is a great idea. It turns out that there are several levels of hints. The hardest hint, but the most useful, is to run through the trick to the place where they think there is 1 penny in each hand, then show them that one hand doesn't have any pennies. There also is a longer story about passing things out evenly, and the importance of going first when there is an odd number of things to be taken.

More Tricks

I gave some students the problem of doing the trick when there are six chickens. Of course, nothing can be done. Nice followup. A farmer had 5 chickens. Everything was fine until he bought two foxes and two wolves. You show two pennies in each hand. Now the farmer leaves and they grab the chickens, the farmer comes back and they put the chickens back, this happens again, and then for a third time the farmer leaves and they take the chickens. The wolves are unhappy, because the foxes got all of the chickens. And you show them one hand with two pennies and the other hand with seven.