Every Sudoku grid hides a coincidence nobody would expect. Take the 16 cells of the ring around the central box and the 16 cells of the four corners, in 2×2 squares. When the grid is solved, the two groups hold exactly the same numbers, with the same repeats.
In blue, the ring; in orange, the corners. The two groups have exactly the same spread: two 1s, three 2s, two 3s, one 4, no 5, two 6s, two 7s, one 8 and three 9s.
It is called the Phistomefel ring, after the puzzle author who made it known. It is not a new rule and it doesn't depend on luck: it follows from the usual rules and holds for any solved grid.
Why it works
Consider rows 3 and 7 and columns 3 and 7. Each holds the numbers 1 to 9, so, putting the four together, every number appears four times. The four cells where the rows and columns cross count twice.
Now consider the corner boxes: 1, 3, 7 and 9. They too have every number four times. So the two groups hold the same numbers.
The grey cells are in both groups. Take each of them once from each group, so both lose the same numbers. The four crossing cells counted twice in the first group, so they stay there once.
The ring is left on one side and the corners on the other. Since the same was taken from both, they still hold the same numbers.
How it helps you solve
The practical rule is this: each number appears in the corners as many times as in the ring. When you know how many times a number can appear in one group, you know how many times it can appear in the other. To find out, count the numbers already placed and the notes.
In this Hard puzzle, the Easy and Medium techniques no longer place any number. The next step would be to look for a naked triple. But look at the ring: its top and bottom rows are full.
Look for the 5 in the ring. It isn't in the full rows. It can't go in the six empty cells either: those on the left are in box 4, which has its 5 in column 1, and those on the right are in box 6, which has it in column 9. The ring has no 5.
So the corners have no 5 either. Three corner cells had the 5 in their notes, and none of them can hold it.
The one in row 8, column 2 only had the 5 and the 8. The 8 stays. From here, the Easy and Medium techniques solve the puzzle to the end.
It also works when the count isn't zero. In this second puzzle, it is counting the 4s that lets you move forward.
The ring already has a 4, in row 6, column 3. Two more ring cells have the 4 in their notes.
In the corners, the 4 can only be in two cells, and both are in column 2. Only one can hold it, so the corners have at most one 4.
So the ring has at most one 4 too, and that one is already there. The other two cells lose the 4. With no hidden triple or X-Wing, the Easy and Medium techniques are enough for the rest.
Is it worth looking for?
The ring doesn't replace the other techniques, and in many grids the count leads nowhere. But it costs little to try when you get stuck, especially when the ring or the corners are nearly full. To count without losing track, paint the ring one colour and the corners another. And even if you never use it, you can always check that it is there in a solved grid.