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# A Perfect Sudoku

I was asked to find a Sudoku solution - a board with all the numbers placed, which had a very specific property. The person emailing me was intending to make a woodwork piece with nine different types of wood arranged in a Sudoku formation. But to make it work he requested every number on the board should be in a different position relative to the 3x3 box each number appears in.

If you look at pretty much any Sudoku solution you will find some numbers that appear twice or more in the same position in a box. I searched through my library of many hundreds of thousands of puzzles and came across just one such puzzle solution. Here it is:
Now, why should this be so rare?

I'm sure it has some interesting mixture of a magic square and a Sudoku. Sudoku solutions have many symmetries so you can still juggle this around and maintain the same properties. The symmetries are:
1. rotation

2. reflection

3. transposing/shifting all of one number to another number, eg swap 1 and 2 etc

4. more interesting...you can swap rows or columns providing the swaps are within the same boxes. Eg you can swap column 1 with column 2 or 3 but not 5,6,7,8 or 9. Same with rows.

5. and lastly, you can swap boxes in groups of 3, eg you can swap the top three rows, with the middle three rows.

That's in case you want to make more versions of the essentially the same board.

You could, for example, reset this solution to be 123456789 along the top row just by substituting 3=1,7=2,9-3,2=4 etc etc for the whole puzzle.

Symmetries don't usually matter for solutions but the principle is the same, and more important, for a puzzle. Any puzzle mixed up by any or all of those symmetries, no matter how many times, will always solve the same way and have the same grade, even though it looks completely different. That's why there are only 4 billion or so possible Sudoku solutions - that are actually different - as opposed to the trillion, trillion ways of arranging them.

Work on this type of puzzle has lead me to create them and there is now a Colour Sudoku solver.

# Comments

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## ... by: Charles Wyatt

The puzzle shown here is not a perfect sudoku puzzle. You've
repeated numbers in the same location in boxes and you've duplicated numbers in the diagonals.

If you would be interested I have such aa puzzIe in a 3x3 9 number, 4x4 16 umber and a 5x5 25 number. I'm working on a 6x6 36 number now.
Andrew Stuart writes:

It could be more perfect I guess, 'perfect' being a bit in the eye of the beholder. To non-duplicate on the diagonal would mean to be looking for a perfect Sudoku X - but why not, it's one less duplication and one is a valid subset of the other. This page was created before I fully understood Color Sudoku and how to make many of them. But I'd happily put up your 9x9 example with credit

## ... by: wHack

Just a quick note. You mention the different symmetries on this page about perfect sudoku grids but the fourth symmetry rule can't be used on a perfect sudoku, as the resulting sudoku will break the additional rule that each number is in a unique location in every box.

## ... by: Rob W

I have been fascinated by this very question for a long time now, and have - after much trial and error - come up with a way of producing "perfect" grids.

However, I tried to take it even further. The example above has lots of "Parallel Pairs" - 4 & 1 in rows 8-9, 8 & 5 in cols 2-3, 6 & 3 in cols 4-5. Obviously, for a solve-able puzzle, one or more of these foursomes have to be provided as starter numbers.

Also tried to remove triplet values (e.g. 8, 6 and 5 in cols 7-8) but found that left far too few permutations to work with ... and caused my laptop to overheat!!!

It really is amazing the tricks you can pick up when you start rummaging.....

Cheers!
Andrew Stuart writes:

This is very true and reminds me of Avoidable Rectangles. There one is conscious of the puzzle makers constraints as well.

## ... by: Nanang

Perhaps I am getting beettr at these, because I enjoyed this puzzle over the samurai star one. It also may have been that I gave myself enough time to do it rather than be interrupted by other things. Looking for the conga-line chains helped.One thing I noticed: I finished the upper right sudoku first and found that even though a part of it was jigsaw that the whole sudoku followed the 3X3 region rule as well. I assumed this was because of the many 3X3 regions in the whole puzzle and shifting between jigsaw and 3X3 for the edges helped me solve the puzzle quicker.
Article created on 21-April-2011. Views: 34045
This page was last modified on 21-April-2011.
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Copyright Andrew Stuart @ Syndicated Puzzles Inc, 2012