## A delicate balance

You are captured by a bandit, but not by an ordinary bandit – an enlightened one.  He will let you go if you solve his brainteaser (otherwise he will keep you and make you solve boring problems every day).  In fact, he will even drive you home in his car.  All you have to do is figure out which one of three keys can be used to turn it on.  The givens: two of the keys weigh the same and the third one is heavier.  He has a balance scale that you can use, but you’re only allowed one weighing.  How will you determine which key is the correct one?

This was how I started this week’s class.  Some of you may recognize this as a reformulation of the classic problem with three coins, one of which is counterfeit.  I had three keys that matched the problem description (it turned out not to be so trivial to find two keys of identical weight among all the teachers) and of course we had a balance scale.  Unfortunately (or fortunately 🙂 ), everyone was so engaged that not a single picture was taken.

In the older group, I was pleasantly surprised when one of the girls solved the problem right away without doing any weightings.  She explained that you need to compare two of the keys and then spelled out how to determine the correct key given all possible outcomes.

In the younger group, the kids also figured out that you need to compare two of the keys but they took some time deciding which two to compare (the keys were actually in bags so that the kids wouldn’t get distracted by their physical appearances).  Once the two chosen ones were weighed, they quickly determined the correct one, but needed a bit of help finding the words for the explanation.  We then discussed how the analysis would change if they had chosen differently.

For homework the kids will get to solve a somewhat harder variation of the problem.  This time the bandit will have nine keys, only one of which will match the car.  The eight wrong keys will all weigh the same, and the correct one will again be heavier.  The kids will have to figure out a way to determine the correct key with two weightings.  Can you do it?

I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.
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### 3 Responses to A delicate balance

1. Natasha says:

I think I have a picture, somewhere.
-Natasha

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2. David says:

When I saw puzzles like the 9-key version as a kid, the solutions were always had you use the results of the first weighing to decide which keys to compare next. It was only years later that I realized that you could decide both sets of comparisons beforehand.

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