# Need help on proceding a paper about estimating numbers of sudoku

I was reading a paper that I found via spiked math (http://spikedmath.com/comics/424-the-numbers-quiz-solutions.png): http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf.

I have problem understanding the subsection Reﬁned permutation and relabelling. Can someone please help?

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It would help if you say what you don't understand. The rearrangement strategy is used before in the paper, trying to reduce the number of configurations that need to be counted. – Ross Millikan Jun 24 '11 at 4:40
I can't see where the "2051 possible B2/B3 pairs" comes from. – Covi Jun 25 '11 at 5:04

## 1 Answer

The idea is to find assignments to the first three rows of the Sudoku that yield the same number of completed sudokus.

For every assignment of the first three rows, we assign a canonical form by first applying the unique permutation on the digits $1,\ldots,9$ such that the first square is $1,2,3;4,5,6;7,8,9$, then rearranging the other 6 columns in some canonical way (see details in the paper). Canonicalization doesn't change the number of completed sudokus.

Other operations that don't change the number of completed sudokus is permutations of the rows or the columns. The idea is to start with a list of all canonicalized first three rows, and identify items which are related by row/column permutation. You do this by going over all items, applying all possible permutations, and canonicalizing; now you can identify the item you started with with the item you ended with.

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One thing I don't understand: how does the "2051 possible B2/B3 pairs" come from? – Covi Jun 25 '11 at 4:39
You run the program, and after all equivalent configurations are identified, you are left with 2051 of them. That's an empirical result. – Yuval Filmus Jun 25 '11 at 23:24