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sudoku mania


I never pay
much attention to the crossword-puzzle page of our regular newspaper DeMorgen. I did notice that they
started a new sort of puzzle a few weeks ago but figured it had to be
some bingo-like stupidity. It wasn’t until last friday that I had a
look at the simple set of rules and I was immediately addicted (as I am
mostly when the rules are simple enough!). One is given a 9×9 grid
filled with numbers from 1 to 9. You have to fill in the full grid
making sure that each number appears just once on each _horizontal
line_, on each _vertical line_ and in each
of the indicated 3×3 subgrids!

It is amazing how quickly one learns
the basic tricks to solve such _sudoku_s. At first, one plays by
the horizontal-vertical rule trying to find forbidden positions for
certain numbers but rapidly one fails to make more progress. Then, it
takes a while before you realize that the empty squares on a given line
in a 3×3 subgrid cannot be filled with any of the numbers already
present in the 3×3 subgrid. Easy enough, but it takes your
sudoku-experience to the next level. Anther simple trick I found useful
it to keep track how many times (from 0 to 9) you have already filled
out a given number. If it is 9, you may as well forget about this number
for elimination purposes and if it is 0 it will be hard to use it.
Optimal numbers to use are those that are already 4 to 6 times on the
board. And so on, and so on.

After having traced all back-copies
of the newspaper I ran out of sudokus but fortunately there is a
neverending (sic!) supply of them on the web. For example, try out the
archive of Daily
Sudoku
, and there are plenty of similar sites as, no doubt, you’ll
find by Googling.

An intruiging fact I learned from my newspaper
is that there are exactly 6,670,903,752,021,072,936,960 different
filled-out Sudoku grids. You then think : this should be easy enough to
prove using some simple combi- and factorials until you give this number
to Mathematica to factor it and find that it is

$2^{20} \\times
3^{8} \\times 5 \\times 7 \\times 27704267971$

and hence has a
pretty big unexplained prime factor! This fact needed clarification, so
a little bit later I found this Sodoku
players forum page
and shortly afterwards an excellent (really
excellent) Wikipedia on
Sudoku
. There is enough material on that page to keep you interested
for a while (e.g. the fact that nxn sudoku is NP-complete).

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