## Circular Sudoku

Everything about Sudoku that doesn't fit in one of the other sections

### Circular Sudoku

Just thought I'd mention a nice variant that imo deserves more attention ...

The page http://www.essex.ac.uk/maths/misc_pages/CircularSudoku.htm has a few links to "circular sudoku"; e.g. a sample puzzle at http://www.essex.ac.uk/maths/misc_pages/Sudoku1.pdf is such that each of the digits 0-9 must occur in each of 5 concentric rings and in each of 10 pairs of adjacent pie-shaped sectors.

Personally, I find the circular layout unappealing (mostly due to the cells not having a fixed size), but a design that's entirely equivalent -- and imo more appealing -- is rectangular with rows that "wrap around" (so the leftmost & rightmost columns are considered adjacent). In such rectangular form, the puzzle equivalent to the one just mentioned is as follows ...
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`Circular Sudoku in rectangular form with wrapping rows:    . . . . . 8 2 . . .    9 8 . . . . . 5 2 .    6 . . . 2 . . 8 9 1    . 3 6 . . 5 . . . .    . . . 4 7 . . . . .`
In this form, the digits 0-9 must occur in each row and in each of the ten pairs of adjacent columns. I enjoy this variant a great deal!
r.e.s.

Posts: 337
Joined: 31 August 2005

### Re: Circular Sudoku

This is just an interleaved pair of 5x5 Latin squares: one square on odd columns, the other on even.

Barring typos, your puzzle is equivalent to the pair:
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`...2.         ..8..9...2         8..5.6.2.9   and   ...81.6...         3.5....7..         .4...`

Ignoring all the symmetries and just doing a straight count, the total number of solution grids for circular sudokus of radius R works out as (2R)! (R-1)!^2 L(R)^2, where L(R) is the number of reduced Latin squares of order R. So, that's:
• 2 of radius 1
• 4 of radius 2
• 2880 of radius 3
• 23224320 of radius 4
• 6554832076800 of radius 5
• 610511815767490560000 of radius 6 ...
... and so on.
Red Ed

Posts: 633
Joined: 06 June 2005

### Re: Circular Sudoku

Red Ed wrote:This is just an interleaved pair of 5x5 Latin squares: one square on odd columns, the other on even. [...]

Indeed. For some reason I mistakenly thought the two squares were not only interleaved, but interdependent (in a way other than merely having the necessary distinct alphabets).

In a futile attempt to redeem the situation, I then thought maybe some interdependence could be introduced by using an odd-sized alphabet, say 1-9, and having strategically-placed cells blocked out as not to be used, like this (where X's mark the cells not to be used):
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`   X . . . . . . . . .   . . X . . . . . . .   . . . . X . . . . .   . . . . . . X . . .   . . . . . . . . X .`

Again the puzzle would use wrap-around, with units consisting of rows and adjacent pairs of columns, but again it reduces to two independent interleaved Latin squares. Oh well.
r.e.s.

Posts: 337
Joined: 31 August 2005

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