# Covering the Bases: Suduko

Today, the “shotgun” passenger in our vanpool was playing Suduko, and it got me to thinking as I was driving. Given that you’ve got a 9×9 grid, there are just so many puzzles. Yes, you get some variation by what clues you could give, but there are still a finite number of puzzles. First question: How many?

Second question, and a more interesting question. Think about Suduko puzzles in different bases. First of all, you can only do it in certain bases, for you need a base that can form a square, and have internal squares. Thus, you could only do it for puzzles that exclude zero for (nx3)+1, or if you include zero, nx3. For example, the current one works because you exclude zero, and thus the puzzle size is (3×3), giving 9 in each row. Similarly, you could do base 7 (but it would be easy), or 6 (if you include 0).

Update: The math was all wrong. The puzzles must be square, and thus the bases would have the progression, if you exclude zero, of n2+1, or 1, 5, 10, 17, 26, 37… If you include 0, you could subtract 1, so a 4 squares of 4 x 4 would be base 16, i.e.,

```0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F
4 5 6 7 | 8 9 A B | C D E F | 0 1 2 3
8 9 A B | C D E F | 0 1 2 3 | 4 5 6 7
C D E F | 0 1 2 3 | 4 5 6 7 | 8 9 A B
- - - - + - - - - + - - - - + - - - -
```

(and so on)

Now, think about higher bases. A 4 square gives you a Base 13 17 Suduko. (hex, if you include 0). Geek Suduko, according to ellipticcurve. Evil. A 5 square, gives base 17 26.

The mind boggles.