Puzzles
Turning squares
Each square on a chessboard contains an arrow point up, down, left or right. You start in the bottom left square. Every second you move one square in the direction shown by the arrow in your square. Just after you move, the arrow on the square you moved from rotates 90° clockwise. If an arrow would take you off the edge of the board, you stay in that square (the arrow will still rotate).
You win the game if you reach the top right square of the chessboard. Can I design a starting arrangement of arrows that will prevent you from winning?
Show answer
Hide answer
No, I can't.
If I could, then my arrangement would cause you to follow an infinitely long pattern without visiting this pattern. As there are only a finite number of squares, within this pattern there must be a square that you visit infinitely often. The arrow on this square will point in each direction an infinite number of times, so you must also visit the squares next to this one infinitely often.
For the same reason, you must visit the squares next to them infinitely often, and the squares next to them, and so on. In this way, we see that you visit every square, including the all-important winning square, infinitely often.
