I’m delighted — and frankly very surprised — to discover a new sudoku rule, to complement yesterday’s Rule of N. And within 24 hours, too, at a point when I was starting to despair. This is very exciting partly because it actually helps solve the Diabolical puzzle that the Rule of N alone wouldn’t solve, but mainly because it returns sudoku-solving to a workable process, without having to resort to guesswork and backtracking.

I’ve called this rule… (fanfare, please)… the Intersection Rule. In true mathematical style we’ll set up some definitions first.

A cell group is a row, column or 3×3 box of nine cells. If we have two cell groups which overlap then the cells which are in both cell groups are the intersecting cells. If we consider just one cell group then the collection of cells which aren’t among the intersecting cells is called the disjoint set of cells. So there are two disjoint sets: one from one cell group, and one from the other cell group. The intersecting cells plus the two disjoint sets together make up all the cells we’re talking about.

Now the Intersection Rule goes like this:

Suppose you have two cell groups with intersecting cells. And suppose further that there is a digit which could go in some of the intersecting cells but which can’t go in any of the cells in one of the disjoint sets. Then that digit cannot appear in any of the cells in the other disjoint set.

This really does need an example, so take a look at this:

Here we have two cell groups (a row and a box), and we’ve coloured the intersecting cells yellow. The two disjoint sets have been left white: there’s one disjoint set of six cells running off to the right, and the other is the six cells at the bottom.

Now let’s consider the row. We can see that the digit 5 can appear in the intersecting cells, but cannot appear in the other (white, disjoint) cells in the row. Now the Intersection Rule tells us that 5 cannot appear in any of the cells in the other disjoint set (the six lower white cells). Specifically we can remove the possibility of a 5 in the middle-middle cell and the middle-right cell of the 3×3 box.

This specific example comes from the Diabolical puzzle mentioned before. And just applying this one instance of the Intersection Rule happened to free it from the stalemate — after this it was possible to apply the Rule of N again repeatedly, and that resulted in a complete solution.

A couple of final notes.

First, when the Intersection Rule talks about “two cell groups” it’s only interesting if one of the cell groups is a 3×3 box. If instead one is a row and one is a column then it’s still true but it doesn’t give you anything beyond what you could have guessed ordinarily.

Second, I still don’t know if the Rule of N plus the Intersection Rule is sufficient. It’s possible that there are sudoku puzzles out there which do have exactly one solution but which cannot be solved with these two rules alone. However I have more interesting things to do than go looking for those.

The main thing is we now have two terrific and useful sudoku rules, neither of which involves guesswork. Doesn’t that restore your faith in the Universe?

Tags: logic, deduction, sudoku