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Math Puzzles

I’ve returned from my conference equipped with a few new ideas, and three interesting mathematical puzzles:

  1. Show that, in any community of finitely many people, at least two people have the same number of friends. (Assume that no one is friend of himself.)
  2. In a certain park, there are some trees, and in those trees are some birds. Each tree has the same number of birds as every other tree. The total number of birds is between 200 and 300. Jane, on learning how many birds were in the park, was able to deduce how many birds were in each tree. How many trees are in the park?
  3. What is the next number in the sequence 1, 11, 21, 1211, …?

Now the humbling part: The first two are questions from math competitions for middle school students.

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