Friday, 28 June 2013

Friday Puzzles #215

Christof S posted a beautiful Klein bottle sudoku earlier this week.  This led me to revisit my own dabbling with sudoku on various square identification spaces.  My own 9x9 version of the Klein bottle was somewhat flawed as although my rows were all of length 9, I had 4 rows of length 18 and one of length 9.  By setting things up as a 10x5 grid Christof has come across a very elegant solution.

Unfortunately I have not even begun to scratch the surface transferring over the images that used to exist on the Warwick servers over to blogger so you (currently) can't see my own Klein bottle puzzle, and nor can you see my interpretation of the real projective plane.  That wasn't very good either in retrospect, bearing in mind there were now four rows and four columns of length 18, and just the one row and column of length 9.

I've decided to rescue this Christof-style by having a much smaller grid.  Correctly interpreted, this grid has three rows, three columns and three regions of area 12.  Rather than ask you to enter in the numbers from 1-12, I'm asking you to enter them in modulo 6.  This ambiguity means I don't need a minimum of 11 clues.  Here I get away with 10, perhaps with more cunningly shaped regions you could have fewer.

That's probably as confusing to most of my Dearest Readers as the notion that the real projective plane is simply the quotient of a regular 2-sphere by the antipodal map, so you can also view the puzzle this way: a standard 6x6 Latin square with 3 regions of area 12 in which each number from 1-6 appears exactly twice.  Given the size, this isn't particularly hard, but it's also in no way as trivial as a standard 6x6 sudoku.  You'd definitely better be sure that exactly two of each number are in each region.  Enjoy!
    #252 Topological Sudoku – rated medium
All puzzles © Tom Collyer 2009-13.

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