Phistomefel Ring Explained & How To Solve Sudokus With It

The Phistomefel Ring (also known as Phistomefel’s Theorem) is the observation of a particular pattern in completed Sudoku puzzles that can aid with solving uncompleted puzzles. 

The observed pattern is that the 16 digits in the four 4×4 corner regions (highlighted by the darker cells below) will match the digits in the 16-cell ring circling the central 3×3 region (highlighted in red below). 

It’s important to note, however, that this does not reveal whereabouts in either region particular digits will appear. Just that the set of the digits in the corners will match the set of the digits in the Phistomefel Ring. 

For example, a look at the completed Sudoku below. 

In this particular example, the set of the digits in the four corner regions is: 2, 6, 4, 7, 9, 5, 8, 2, 4, 1, 7, 3, 8, 2, 1, and 5. 

This perfectly matches the digits in the Phistomefel Ring. 

You’ll notice that there are three 2s in the corner regions and there are exactly three 2s in the ring. 

The digit 9 only appears once in the corner regions and, as stated by the Phistomefel Theorem, 9 only appears once in the ring. 

How to use the Phistomefel Ring to help solve Sudokus

The Phistomofel Ring is not just an interesting example of set theory, but can also be used as a solving technique to help you crack puzzles. 

For example, you can think of this as an additional constraint to the usual constraints of Sudoku. 

So if, for instance, you have solved all 16 cells of one of the particular regions, you immediately know what digits appear in the other region and how often they appear.