Puzzles Michaelmas Term 2012

  1. There are three men in the room: one of them always tells the truth, one always lies and one answers yes/no randomly. Each of them knows who is who. What three yes/no questions would you ask them to determine who is who?
  2. Given n different subsets of the set of numbers 1,2,3,…,n, prove that there exists a number, such that if it is left out of each subset, the remaining subsets will still be different.
  3. A town is surrounded by a circular wall. There are 12 guards serving on the wall. At twelve noon, each guard leaves his watchpost and starts walking the wall in some direction, at a speed at which it would take exactly one hour to walk around the whole town. If two guards meet, they both turn around immediately and walk at the same speed in the opposite direction. Prove that at twelve midnight each guard will be back at his watchpost.
  4. Prove that there exists a natural number n satisfying following properties: it has at least 100 digits in decimal representation, it is not divisible by 10 (so far so good) and it is possible to interchange two different digits of n to obtain a number with the same set of prime factors as n.
  5. A jigsaw puzzle has tiles that can be assembled to form a 23 x 37 rectangle. The pieces come in five shapes, as shown below. Can you say how many pink pieces must be included in the puzzle?
  1. An n x m rectangle is composed out of nm unit squares (n and m are integers). Draw a diagonal of the rectangle. How many unit squares does it go though?

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