Logic Puzzles

  1. An air squadron has about 50 planes. It's flight pattern is an equilateral triangle; every plane except the first is halfway between two planes ahead of it. For this problem, an equalateral triangle must have at least 3 planes. Several planes are shot down in combat. When the squadron returns, the planes form four equilateral triangles. The lost planes could have formed another equilateral triangle. If all these triangles are different in size and how many planes were there to begin with?
  2. Timothy wants to saw a cube of wood into 27 equal cubes. During the work, if several pieces are already cut, he can arrange them as he pleases, and saw through all of them with one cut. Working this way, how many operations are needed?
  3. 10 coins are in a row, five heads on the right and five tails on the left. In as few moves as possible, we want to alternate heads and tails. The only move permitted takes two consecutive coins and places them in a two coin wide space in the same order they were picked up in. If there is no space between coins, the two coins can be placed at one end of the row. For example, here are the first and second moves of one attempt
    HHHHHTTTTT
    HHHH..TTTTHT
    TTHHHH....TTHT