Brain Teasers

Knight and Coins

Bob and Jane are taking turns, placing knights and coins respectively on a chessboard. If Bob is allowed to place a knight only on an empty square which is not attacked by another knight, how many pieces at most can he place before running out of moves? Assume that Jane starts second and plays optimally, trying to prevent Bob from placing knights on the board.

Bob can place at most 16 knights. One way to do this is to keep placing knights only on the 32 white squares. In order to see that Jane can prevent Bob from placing more than 16 knights, split the board in four 4x4 grids. Then, group the squares in each grid in pairs, as shown on the image below. If Bob places a knight on any square, then Jane will place a coin on its paired square. This way Bob can place at most one knight on each of the four red squares, one knight on each of the four green squares, one knight on each of the four brown squares, and one knight on each of the four blue squares. Therefore, he can not place more than 64/4 = 16 knights on the board.

Two Missiles

Two missiles which are 2000 miles apart are shot towards each other. The speed of the first missile is 13000 miles per hour and the speed of the second missile is 17000 miles per hour. Find the distance between the two missiles 2 minutes before they collide.

The distance will be 1000 miles. It does not matter what he starting distance between the missiles is. 2 minutes before collision they will be (13000 + 17000)/30 = 1000 miles apart.