Some Fun Games and Puzzles

Abstract

“Set-the family game of visual perception” is a card game by M. J. Falco containing 81 cards. It can be played both as a solitaire and as a multiplayer game and is highly addictive. The game has a very nice description in terms of finite geometry. Here are the rules: The 81 cards are the points of the four-dimensional affine space over the field with 3 elements. The dealer shuffles the cards/points and lays twelve of them (in a rectangle) face up on the table so that all the players/geometers can see them. The geometers try to spot sets of three points among these 12 points that are lines in our four-dimensional space. Whoever spots a line first gets to keep the three points and replaces them by three points from the stack of unused points. Since there are arcs of 12 points in our space, that is, sets of 12 points in our space that do not contain any line, it is possible to get stuck at this point. If this happens, three more points (making a total of 15) are laid up. There is always a line to be found among 15 cards (this means that maximal arcs in our space have fewer than 15 points). After removing a set of three points of a line, these are not replaced, and the number of cards is reduced to twelve again. Back to step one. Continue until all the points of the space are used up and finish the game by removing all lines from the remaining set of 12 cards. The geometer who collects the most points wins the game. It turns out that it is always possible to find a line among the last twelve points and that the game ends with either 0, 6, or 9 points left on the table.