Andreas Nortmann wrote:
I have to take a second look into that article, since that is Oscars second puzzle based on it.
Implementation of groups on puzzles has always been an interesting concept,
especially as most of the time it is the other way around.
Now, implementing simple groups (i.e. the building blocks of group theory), especially the
sporadic ones (which are "one of a kind" with the complete sense of those words!) may add more
complexity to the challenge. And after all, all groups can be written as permutation groups,
and a solution will always be there, but it just gets tougher to solve a puzzle intuitively.
Oskar has made two different puzzles which represent the M12 group, which has two main
generators, one along the equator, and another permutation which involves less terms.
The real challenge though, would be to make one-piece of such puzzles which will prevent
cheating. For example, if in the Number Planet, the "equator move" is allowed without having
all pieces on one side, then it is easy to reset the puzzle, but also to cheat.
(I am not sure if there is an "aligning trick" to prevent this, but if ever I play with it, then I'll know!)
But maybe, allowing the "cheating mode" could help the puzzle become easier for the mass,
while always hiding that great "sporadic power" for the real mathematical experts.
Of course, there are much more ways to make such puzzles. If I had to plan a similar implementation,
I would use some "rotating tunnels". That could be useful, for example, for other simple groups too
(like the M24 and the "Dotto" group).
Those puzzles may not have much appeal to the mass market, but I can imagine that if a puzzle
is flexible enough to "change" the number of elements and its "generator tunnels", i.e. to be able
to simulate many different groups, then it would be interesting and useful for every University's
Pure Mathematics Branch.
Here is a nice link to view and play M12, M24, and the Dotto groups:http://www.neverendingbooks.org/index.p ... games.html
But the theory behind all these is simply wonderful, I would encourage everyone to study more
the Atlas of Finite Group Representations. It is a perfect world inside our imperfect real world.