The game of CON-TAC-TIX occurred to Piet Hein while he was contemplating the famous four-colour theorem of topology. The theorem, as yet unproved, it that four colours are sufficient to make any map so that no two countries of the same colour have a common boundary. Piet Hein introduced the game in 1942 at the Niels Bohr Institute. The same year a leading Danish newspaper published an account of the game; it soon became enormously popular in Denmark (under the name of Polygon). Pads on which the game could be played with a pencil were sold, and for many months the newspaper ran a series of CON-TAC-TIX problems, with prizes for the best solutions. Albert Einstein who was passionately interested in games had since 1948 Piet Hein's CON-TAC-TIX on a shelf in his study. ----------- CON-TAC-TIX is played on a diamond-shaped board by two players. Two opposite sides of the diamond are marked black; the other two sides are white. The holes at the corners of the diamond belong to either side. One player has a supply of black pins; the other, a supply of white pins. The players alternately place one of their pins in any one of the holes, provided the hole is not already occupied by another pin. The objective of Black is to complete an unbroken chain of black pins between the two black sides. White tries to complete a similar chain of white pins between the white sides. The chain may freely twist and turn. See example of winning chain. The players continue placing their pins until one of them has made a complete chain. The game cannot end in a draw, because one player can block the other only by completing his own chain. These rules are simple yet CON-TAC-TIX is a game of surprising mathematical subtlety.