Merlin's Magic Square

Merlin is an electronic game, manufactured by Parker Brothers, which plays six different games - Tic Tac Toe, Music Machine, Echo, Blackjack 13, Magic Square and Mindbender. We are only concerned with Magic Square here as it is a Lights Out type game. Magic Square is played on a 3 by 3 grid of buttons which can take one of two states, off or flashing. When the player presses one of these buttons, the state of that button and of certain of its neighbours is toggled. The aim of the game is to get all the buttons flashing except the central one. The rule that determines which buttons are affected by any particular button press is described below.

If you would like to know more about the other Merlin games you should visit the Virtual Merlin pages. There you will find an astonishing amount of information (including the original manual as a PDF file) and also a Merlin Simulator for Windows, which plays all the games. This is well worth downloading.

Another interesting reference for Merlin's Magic Square is a paper by Don Pelletier called "Merlins Magic Square" ( American Mathematical Monthly , 94 (1987) 143-150).

The rule

This is best described in pictures. In each of the diagrams below, pressing the red button will alter the state of itself and of the yellow buttons

Although these rules are quite different from the rules for Lights Out, they are still very simple and easy to remember. They do make for quite a good puzzle and you can have a go by playing my Javascript Merlin game. Merlin's Magic Square is a straight simulation of the SIGMA+ game . You can see the graph that is used here .

Analysis of Merlin's Magic Square

The mathematics of Merlin is very similar to that of Lights Out, and I'll be writing some pages about it soon. In the mean time I suggest that you read the paper cited above if you have access to a library. Some key points are

Every possible initial configuration of lights is solvable
Merlin chooses a initial configuration for the buttons at random, so this is good news. Notice that this is not true for Lights Out (at least for standard Lights Out played on a 5x5 grid)

The maximum number of moves needed to solve a puzzle is 9
Indeed the pattern shown below requires that you press all nine buttons to solve it (there is no shorter method)

Merlin Links

Links to other information about Merlin on the web

Virtual Merlin	As mentioned above, these pages have a huge amount of informastion about Merlin, including a downloadable simulation for Windows (which is pretty darn good by the way!)
Camelot Software	Another simulation of Merlin for Windows. I haven't tested this one yet, but it looks like it should be worth a go.
Merlin's Magic Squares	An article about Merlin's Magic Square, including some information about the underlying mathematics, and also a (rather ugly) Java applet that allows you to play Merlin's Magic Square with 2,3 or 4 states.