THEOREM: A cat has nine tails.
PROOF: 1. No cat has 8 tails.
2. One cat has one more tail than no cats. Therefore, a cat has nine tails.1
(May 14, 2013) We began today’s Math Circle debating the merits of the cat-has-nine-tails “proof.” N stood at the board diagramming “my” reasoning. G came up to the board, picked up the chalk, explained her position, and began calling on people for their opinions. (“This is fun,” she said to me in an aside.) I sat down, put my feet up, and enjoyed watching the debate. The class had things well at hand and didn’t need me.
A group instinct emerged that in the proof, some sort of semantic shift occurs between the two steps. They attacked my reasoning by picking apart the words “no cat,” “one cat,” and “has.” I, playing the stooge, held my ground by restating the proof using imaginary visuals. The kids didn’t buy it. But they disagreed with each other about the proof’s validity. C claimed that while a semantic shift does occur, the proof is still valid. “It makes perfect sense to me,” she asserted. N concurred.
G, A, and P disagreed. “I feel that there is a problem with the wording that makes the proof not valid somehow; I’m just not sure what it is,” explained A.
“This would be a lot easier if you would just tell us the answer,” said G, addressing me. I promised that next week, after the students have had a week to think about it, I’ll offer an opinion. Just then, R, who was late, walked in. We moved our attention to the continuation of last week’s irrationality of root 2 proof. I reminded everyone that this proof is by contradiction.
“If you are purposely manipulating the proof to get a contradiction,” declared R, “then it doesn’t hold up logically. It would only be valid mathematically if you stumbled upon the contradiction accidentally.” R’s comment had revealed a gap in my knowledge. (Don’t you hate when that happens?) I know how to do proofs by contradiction, and I have heard it said that not everyone likes them, but I actually didn’t know the precise reason for these objections.
Actually, I use the phrase don’t you hate when that happens a bit facetiously. I actually like when that happens. Yes, my heart skips a beat and I may think “uh-oh,” but for me, these instances are great opportunities to remind, or inform, kids that teachers do not know everything. We need to know enough to explore interesting questions with kids, and we need to be curious enough to get excited about a chance to learn something new. At my first Math Circle training with the Kaplans, Bob and Ellen gave the advice to “choose a topic that you know something, but not too much, about. During this course, I have been excited about these kids’ challenging questions about the Pythagoreans. Every week I race home to do more research, and then come into class the next week to say “I can’t wait to tell you what I learned this week about Pythagoras!” Curiosity is contagious and gets everyone more excited to learn.
The irrationality proof that we’re working on emerged from the kids’ curiosity. It was not, as I have mentioned, in my plans. In retrospect, I’m uncertain: was this an ideal proof to engage in so soon? The algebra has been hard for the kids – really hard. And the end result, once we muddled through all that algebra, was not very satisfying. “That’s it?!” complained a few kids upon arrival the punch line.2
They complained that the proof was stilted, manipulated, and just not obvious. I explained that mathematicians are constantly trying to make proofs more elegant. An obvious (in hindsight) proof is a beautiful proof. No one found this proof beautiful.
“It sounds like you’d all like something more direct,” I said. “Here’s something you can prove directly.” I drew a big X on the board and asked kids to name the angles.
“Picard,” said G.
“Honeybadger,” said N.
“Pananca,” said G.
“x,” said A.
“Now,” I instructed, ”prove to me that the measure of angle Picard is equal to the measure of angle Honeybadger.”
“They look like they’re all equal,” said R.
“You can’t go by how they look. They might not be equal.” That statement was met by silence, so I asked students what they knew. They knew that supplementary angles add up to 180 degrees. We listed all four supplementary combinations on the board, and looked. And looked. And looked.
“I think that it is somehow implied that those angles are equal,” said R.
“An implication is not a proof, though. You need to make it obvious,” I replied. At this point, sadly, we were out of time. I sent the kids home thinking about this proof and about cats with nine tails, to be continued next week.
1 This proof is all over the internet, so I’m not going to list a single reference. Just google “cat has nine tails proof” when you’re looking for a fun diversion.
2 From a pedagogical perspective, I’m thinking that an elegant, satisfying proof with challenging algebra would work well in this Math Circle, as would its counterpart: a proof with easier mathematical manipulations but a flavorless ending. Here, though, we had a tough mathematical exercise without that great feeling of accomplishment that comes from a satisfying proof. This proof had its benefits, though. The kids learned a lot of arithmetic and algebra, they loved the history surrounding it, and it got them to realize that not all proofs are of a mold.