# A Sudoku Surprise

Once upon a time, I was a sudoku fiend. They provided an outlet, a distraction, a hiding place – I could bury my head in one for, say, half an hour and whatever had been troubling me before was somehow less of an issue.

It’s not an effective *strategy* for dealing with day-to-day living, but it was a useful tactic. (I remember being advised, at one point, that there wasn’t a global backlog I had to help sort out.)

And at some point, I got bored of them, or needed to hide less, and got out of the habit. I’m still interested in them mathematically, and when someone posted one on reddit with a question about how to solve it without guessing. It turned out, it sort of can - although it involves a fairly sophisticated shape I hadn’t seen used before.

Here’s the grid - have a go, if you’d like to – there are spoilers below the line

The critical shape is in the two big squares in the bottom right, and it’s made up of four pairs of cells: in the bottom-middle square, the 12 pair and the 37 pair; in the bottom-right square, the 13 and 27.

There’s one extra bit of information I need to use: because of the 17 in the bottom left corner, one of the cells in the bottom row *must* be a 3.

That means the 37 in the middle line and the 13 in the top line can’t *both* be 3. (One of them must, however).

Now, let’s look at the 12 and 27 pair, which are arranged so that the 2s must be either in the top-left and bottom-right or top-right and bottom-left. BUT, in that second case, the opposite corners must be a 7 in the bottom-right and a 1 in the top-left - but that makes the middle 37 and top 13 both threes, which we can’t have!

So, in this scenario, the 1 must be below the 2 and the 7 must be above it. This situation gives no (direct) information about the two remaining pairs, although the rest of the grid falls apart once the 2s are in place.

### Remaining questions

I have some questions about this.

- How do we spot such a pattern?
- How do we notate the grid so that such patterns are obvious?
- Is there a more succinct explanation of why the pattern works as it does?

I’d love to hear your thoughts in the comments!