Archive for June 2010
First, a couple of puzzles, almost but not entirely unrelated to the subject of this post, just to warm you up.
Cut an obtuse triangle into acute triangles. *Prove that 7 is the smallest number of pieces.
Cut a square into acute triangles. *Prove that 8 is the smallest number of pieces.
And now, the real problem.
A square can not be cut into an odd number of equal area triangles. (Note that this problem deals with area, unlike the warm-up puzzles that I offered).
I decided to write about this problem because I see it pop up more and more often in internet forums. It looks like people don’t lose hope that somebody may discover an elementary solution. Well I am very skeptical about this. So I want to offer a sketch of the proof that was published in 1970 by Monsky. The proof is not elementary, but it is beautiful. To me, it is like a game of chess (I don’t play chess) – I would never be able to come up with anything like this, I can not learn from this, but can appreciate all the strategy and tactics employed.
You write down 2 distinct real numbers from [0,1] and seal them in envelopes. I pick one envelope, open it, and guess whether the number that I see is greater or less than the number in the other envelope.
I claim that I have a winning strategy, i.e. a strategy that allows me to guess correctly with probability greater than 50%.
Such strategy does exist.
It is important to note that you don’t pick numbers “at random”, you pick them “at will”. That is, I cannot assume that they obey some certain distribution. I intend to win no matter how you try to counter my strategy.
There has been little doubt that they do oscillate, but still…
One cool thing about this experiment is that neutrinos produced in it are more energetic than those from any known natural sources. It is often erroneously claimed that the LHC will produce “unprecedented, never seen in the Universe energies”. Well, the OPERA experiment has apparently been creating neutrinos of unprecedented energies.
Neutrino is an anagram of “inner out”