Fairy Tale Math Circle #5

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We took a long break because of the 4th of July holiday, but we’re now back. This past Monday’s fairy tale was Крошечка-Хаврошечка – a yet-another strange tale with a hard-working orphan girl, the three lazy sisters, a cruel stepmom, and a prince on a white horse. Reminds you of something? Well, the kids thought it was very similar to the Cinderella story except for the fairy godmother. I asked if the talking cow that helps the girl and eventually turns into an apple tree can be counted as a sort-of fairy godmother, but they felt it was too much of a stretch. Anyway…

While we were waiting for everyone to arrive, we played with paper and a single-hole hole punch. You see, in this fairy tale, the first mean sister has just one eye; the second sister – two eyes; and the third sister – three eyes. So we started by figuring out how to punch enough eyes with just one punch. That led to a pretty lengthy investigation into many-eyed monsters that could be created by folding a certain way and then punching just once. Creating a 2- and a 4-eyed ones was easy. But what about a 6-eyed one? How about a 5-eyed creature? A 10-eyed? 40-eyed?

After that the kids worked on a problem of three sisters’ three trinket boxes. The sisters had their names written on the boxes, but then in order to confuse the poor little orphan girl they changed the names around so that none of the boxes had correct names on them.

My science experiment for the day was actually another math puzzle – How to walk through a sheet of paper. In the fairy tale, the orphan girl routinely climbs into the cow’s one ear and out of the other. Everyone got busy cutting big rectangular holes in sheets of paper so they looked like picture frames. N (new to the group) suggested that a small child could fit through such “frame”. But when we tried this with L’s two-year old brother, the frame didn’t go past his shoulders.

Now, M came up with a sideways solution. I used the Russian word через when posing the problem. It has two meanings – through and over. M decided to use the later and brought me carefully-folded paper to “step over”. This led to a short discussion of how words can be imprecise and about word-play.

After a while M and N came very close to solving the problem. We analyzed their solutions (with scissors in hand) and they figured out the missing piece of the puzzle.

At this point I decided to give them the following little problem:

The evil stepmom sends the little orphan girl to the apple tree to pick apples. And she tells her to place collected apples into three baskets in such a way that the total number of apples in any two baskets was always an odd number.

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All five of the kids got working on the problem. N’s first suggestion was to put zero apples into each basket. But V and D objected that zero was an even number. So we quickly reviewed what made numbers odd or even. Next, N and L paired up and started coming up with solutions involving negative apples. They presented at least a dozen ideas, but on closer examination none satisfied the condition.

V and D also paired up. They stuck to positive numbers only and were very careful about checking their solutions before calling me over. Finally, D declared that the problem was impossible to solve.

M ended up working by himself. He tried 6, 7 and 5 apples. It seemed to work, but then he checked his arithmetic (7+15 was NOT 13). He then sat there thinking for a while until he presented a solution that actually worked. He suggested placing half an apple into each basket. This was unexpected and thrilling. So we discussed why it was not possible to solve this problem using whole numbers. Then D wanted to know whether there were some numbers that were both even and odd at the same time. Sadly, this was at the end of the circle. I guess we’ll have to come back to this question next time we meet.

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