I'm leaving for a two-week holiday tomorrow, so thought to leave some questions I've wondered about and that the mathematicians among you might have an opinion about (like: "He's nuts!").

Imagine a hyperspace of 81 dimensions. Each axis extends from 0 to 9. Take any minimal, valid sudoku puzzle in its canonical form. It can be represented as a point in this sudoku space.

I have programs that fiddle around with puzzles: changing values, deleting cells, binding in a new cell so that the result becomes minimal. When I observe these programs working, I notice that some puzzles can be readily and quickly transformed into a multitude of others. These new variations (once isomorphs have been deleted), are neighbours of the original in sudoku space. Other puzzles are difficult, if not impossible, to vary, and those with few variations often take long to process in this fashion. Such puzzles appear isolated in sudoku space.

Now to the questions. Our universe consists of galaxies, clusters of galaxies etc. Does the set of valid sudoku puzzles also form shapes, hyperplanes or branes in sudoku space? If so, could these be identified using, say, cluster analysis or principal-components analysis? And how, if at all, might that be linked to 'difficulty'?

Any comments are most welcome.

Regards,

Mike Metcalf