Puzzles: Simple Groups at Play

This story is a supplement to the feature "Rubik's Cube Inspired Puzzles Demonstrate Math's "Simple Groups" which was printed in the July 2008 issue of Scientific American.

M₁₂ Puzzle
Mathieu puzzle M₁₂, which represents the sporadic simple group M₁₂, was designed by the authors to be played on the Internet. The puzzle begins with a scrambled version of the numbers 1 through 12. The object is to unscramble them using combinations of just two moves, both conveniently executed at the click of a button. The diagram shows the effect of each move on the unscrambled numbers.

Play the M₁₂ Puzzle online (Note: Puzzle solution can be found here)

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M₂₄ Puzzle
A second Mathieu puzzle, M₂₄, represents the sporadic simple group M₂₄. In the unscrambled state the numbers 1 through 23 are arranged in a clocklike circle, and a 0 is placed just outside the circle at 12 o’clock. As with the M₁₂ puzzle, the object is to restore the unscrambled order from a scrambled state. The M₂₄ puzzle also has two moves. One move rotates the circle one “notch,” sending the number in position 1 to position 2, the number in position 2 to 3, and so forth. The number in position 23 is sent to position 1, and the number outside the circle does not move. The second move simply switches the pairs of numbers that occupy circles having the same color.

Play the M₂₄ Puzzle online

Dotto:
Dotto, our final puzzle, represents the Conway group Co₀, published in 1968 by mathematician John H. Conway of Princeton University. Co₀ contains the sporadic simple group Co₁ and has exactly twice as many members as Co₁. Conway is too modest to name Co₀ after himself, so he denotes the group “.0” (hence the pronunciation “dotto”).

In Dotto, there are four moves. This puzzle includes the M₂₄ puzzle. Look at the yellow/blue row in the bottom. This is, in fact, M₂₄, but the numbers are arranged in a row instead of a circle. The R move is the "circle rotation to the right": the column above the number 0 stays put, but the column above the number 1 moves to the column over the number 2 etc. up to the column over the number 23, which moves to the column over the number 1. You may also click on a column number and then on another column number in the bottom row, and the "circle rotation" moving the first column to the second occurs. The M move is the switch, in each group of 4 columns separated by vertical lines (called tetrads) the "yellow" columns switch and the "blue" columns switch. The sign change move (S) changes signs of the first 8 columns (first two tetrads). The tetrad move (T) is the most complicated: Subtract in each row from each tetrad 1/2 times the sum of the numbers in that tetrad. Then in addition to that, reverse the signs of the columns in the first tetrad.

Strategy hints: Notice that the sum of squares of the numbers in each row doesn't change. (This sum of squares is 64 in the first row, 32 in every other row.) If you manage to get an "8"in the first row, you have almost reduced the game to M₂₄ except those signs. To have the original position, signs of all numbers on the diagonal must be +. Hint on signs: if the only thing wrong are signs on the diagonal, and only 8 signs are wrong, those 8 columns can be moved to the first 8 columns by using only the M₂₄ moves (M,R).

Download Dotto (Note: This link contains a zipped .exe file and can only be played on Windows computers.)

This puzzle project originated in an NSF "Research experience for undergraduates" project at The University of Michigan.