Problem: find a winning strategy for the game of PenHex.

    The game of Hex is played on a diamond-shaped arrangement of hexagonal (six-sided) tiles. Players take turns placing markers on the tiles with the goal of connecting two opposite sides of the board with a chain of tiles. You can find an interactive version on the internet at www.mazeworks.com/hex7.

    Suppose that instead of hexagonal tiles we used the shapes invented by the mathematician Roger Penrose. Penrose tiles will fit together to cover a flat surface without creating any repeating patterns. Mathematicians call this an "aperiodic tiling" of a plane surface.

    The accompanying diagram shows my first attempt at a PenHex board. The rules are similar to those in Hex, except that you can connect two tiles only if they meet along an edge -- not at a single point. On the left-hand board, player R has made the winning play, and player Y has made a desperate, illegal move. Readers are encouraged to try the game, perhaps making up special rules. For example, you might say that the first player cannot chose the symmetric star-shaped tile.

    Hex was invented independently;  in 1942 by Piet Hein in Denmark, where it was called "Polygon"  and in 1948 by John Nash in the United States, where it was called "Nash." In 1952 the United States company Parker Brothers marked a commercial version of the game as "Hex."

    A description of Nash's version of the game can be found in the book "The Essential John Nash." Roger Penrose has published many technical and popular books. His most recent is "The Road to Reality : A Complete Guide to the Laws of the Universe." Piet Hein wrote many short poems, which he called "Grooks." Here is an appropriate example for the student trying to find a winning strategy for PenHex:

    PROBLEMS
    Problems worthy
    of attack
    prove their worth
    by hitting back.

A blank board is included below:

Created by Mathematica  (February 13, 2006)