The Phistomefel ring is the 16-cell frame immediately surrounding the central 
box in a standard 
Sudoku grid. In every completed
valid grid, the multiset of digits
in this frame equals the multiset
in the four 
corner blocks. Equality includes the number of occurrences
of each digit, rather than merely the sum
of the entries (Phistomefel 2020).
Number the rows and columns from 1 through 9. The corner blocks consist of cells whose row and column indices both belong to 
. The ring consists of the cells with both indices
between 3 and 7 and with at least one index equal to 3
or 7. Each region contains 16 cells.
Counting each digit in two ways proves the identity. Fix a digit 
, and let 
count its occurrences outside the central 
square. Counting 
in rows 1, 2, 8, and 9 and columns 1, 2, 8, and 9 gives eight
occurrences, with each corner-block occurrence counted twice. If 
counts occurrences in the corner blocks, then 
. Alternatively, the eight noncentral 
boxes contain eight occurrences of 
. Their cells are exactly those outside the central 
square together with the
ring. If 
counts occurrences in the ring, then 
. Hence 
for every digit 
(Phistomefel 2020).
The identity follows from the usual row, column, and box conditions of Sudoku; it imposes no additional rule and does not require the puzzle to have a unique completion.
