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!

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

## David Eppstein

That’s totally not how I would have solved this from here. I would look at the cells that could still be 6 or 8. You have two in the bottom right, two in the top left, and four in the bottom left. But the two that we end up choosing in the bottom left must be diagonal from each other (horizontal and they would knock out one of the two in the bottom right; vertical and they would knock out one of the two in the top left), which leaves only two possibilities: R7C1-R8C3 and R8C2-R7C3. And R7C1-R8C3 doesn’t work because it forms a self-contained subset of six cells, that could be filled with the 6 and 8 digits in two different consistent ways, and we know that every sudoku must have a unique solution. So the only choice is R8C2-R7C3 and as R8C2 cannot be a 6 it must be an 8.

## Colin

Oh, that’s very neat! I hadn’t come across that self-contained subset trick, I love it. Thank you 🙂