# Puzzles: Simple Groups at Play

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#### The cycle notation and M12

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)

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.

View
1. 1. kogorman 01:57 AM 6/21/08

The link to dotto.zip is broken. Otherwise, a great article.

2. 2. kickabear 04:13 PM 6/21/08

PC is defined as "personal computer", not "Windows". The Mac is a PC, as it this Linux laptop on which I'm typing this.

3. 3. MIMIMIMIMIMIMIMIMIMI 06:44 PM 6/21/08

M12 solution below.

--
Edited by MIMIMIMIMIMIMIMIMIMI at 06/21/2008 2:14 PM

4. 4. MIMIMIMIMIMIMIMIMIMI 07:06 PM 6/21/08

My name.

--
Edited by MIMIMIMIMIMIMIMIMIMI at 06/21/2008 2:14 PM

5. 5. phatt1 03:28 AM 7/7/08

M12 was fun took 4 hours the 1st time but now on to M24

6. 6. masimonson 05:27 PM 7/9/08

Fun puzzles, I like the cuber better though.

7. 7. maximlagerweij in reply to MIMIMIMIMIMIMIMIMIMI 01:57 PM 7/29/08

So, the solution is bold and simple. It seems a general and long one. By calculation it can be done quicker. Just as Rubiks` cube could be solved by mathematics in 20 steps and with a simple easy to remember solution in 100 steps.

8. 8. jp in reply to MIMIMIMIMIMIMIMIMIMI 04:19 PM 7/30/08

A simple sequence of moves which returns the puzzle to its starting point is interesting but not a general solution. A general solution is an algorithm which returns the puzzle to the ascending 1 to 12 state from any random position. I think the whole point about these puzzles is that (unlike the Rubik Cube) any move tends to jumble everything so it is hard for a human mind to take a systematic approach which visibly gets closer to the solution. It might be that you essentially have to memorise a large number of positions, or "cheat" and look up a table.

9. 9. maximlagerweij in reply to jp 12:53 PM 7/31/08

You are completely right. I drew my conclusion too soon. I tried mimimi etc as a starting point on a randomization and it worked. I assummed that this was no coincidence and that the comment gave the general solution. After your comment I tried again and it did not work. What the percentage of randomizations it will solve and when (after different numbers of repetitions) I did not systematically pursue. I also agree on the Rubik`s cube. It is a matter of memorizing a large number of positions. Another reason why these puzzles are so intriguing is that the seem simple and that anybody assumes that the solution must be simple.

10. 10. harmanjd 07:16 PM 8/8/08

http://www.sciam.com/article.cfm?id=puzzles-simple-groups-at-play

is the link that has the games.

11. 11. steven_elliott 06:40 PM 8/23/08

I had so much fun with the M12 puzzle on the SciAm website that I wrote my own. My version of the puzzle is similar with two main differences:
1) It can solve the puzzle when "cheat" mode is engaged by making use of an internal database that includes all 95,040 positions.
2) It's a Java applet instead of a Shockwave application.
http://selliott.org/puzzles/m12
It's also open source (for those curious how I did it). Feel free to send me any feedback.

12. 12. steven_elliott in reply to steven_elliott 10:05 PM 8/23/08

I didn't mean to post the above 3 times. I guess I got confused by it being on page 2. When I posted it the website flickered something for a fraction of a second, which I thought might have been an error message, and then continued to show page 1. If an admin could delete all but 1 of my posts above that would be great.

13. 13. math618 in reply to 11:50 PM 8/25/08

http://selliott.org/puzzles/m12 cannot open!

14. 14. steven_elliott in reply to math618 01:30 AM 8/26/08

Do you mean it failed with a simple 404 problem, or some other problem? Sometimes my hosting provider is a bit slow, but I copied precisely what you have above into my browser at it worked for me.

If you mean that you can see the page but you're having some trouble with actually running it it should display an error message box indicating if you have the wrong version of Java (before 1.6). If you don't have Java installed (specifically, in your browser) at all you'll just see a gray rectangle. Hopefully you'll be able to get it to work.

15. 15. Annie 10:56 AM 8/28/08

The link from steven_elliott worked for me...
http://selliott.org/puzzles/m12
Nice - thanks.

16. 16. MATH 04:29 AM 6/2/10

MI10
M2I.11
M3I.6
M4I.5
M5I.11
M6I.11
M7I.5
M8I.6
M9I.11
M10I.10
IT IS SYMMETRY

17. 17. MATH 04:35 AM 6/2/10

M1.10
M2I.11
M3I.6
M4I.5
M5I.11
M6I.11
M7I.5
M8I.6
M9I.11
M10I.10
SYMMETRY!
KEEP TRYING

18. 18. MATH 04:38 AM 6/2/10

CORRECT
M1I.10

19. 19. MATH 04:44 AM 6/2/10

M24:
RS.3
R2S.23
R3S.12
,ETC
DOTTO:
MR.3
TS.2

20. 20. MATH 09:27 AM 6/2/10

MI.10means:MIMIMIMIMIMIMIMIMIMI
,ETC

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