A Sudoku Secret to Blow Your Mind - Numberphile

Main ideas and concepts

• The video discusses a little-known but consistent property of all valid Sudoku puzzles called the Phistomefel Ring.
• The property is explained using set equivalence theory: different groups of cells in a Sudoku can be described as containing the same “collection of digits” (as a multiset), even when you don’t know which exact digits are in which individual cells.
• The goal is to show a geometric/deductive pattern: after a specific kind of “removal,” the digits in a ring around the center of the Sudoku match the digits found in the corners.

Key Sudoku rule being used

A standard Sudoku grid requires placing numbers 1–9 so that:

• Each row contains 1–9 exactly once,
• Each column contains 1–9 exactly once,
• Each 3×3 box contains 1–9 exactly once.

Methodology: How the Phistomefel Ring property is derived (step-by-step)

1) Define two equivalent cell sets

The proof defines two groups of cells that are each guaranteed to contain the same aggregate multiset of digits.

• Red set (4 digit-complete sets of 1–9):
  • One entire row (row 3 in the explanation),
  • Another entire row (row 7),
  • One entire 3×3 box,
  • Another entire 3×3 box (e.g., “box 6” in the explanation).

Together, these four regions collectively contain exactly four complete sets of digits 1–9 (one set per region).

• Green set (also 4 digit-complete sets of 1–9):
  • One entire column (column 1),
  • Another entire column (column 2),
  • Column 8,
  • Column 9.

Together, these four columns also collectively contain exactly four complete sets of digits 1–9.

2) Establish equivalence

At this point:

• The multiset of digits contained by the red cells and
• the multiset of digits contained by the green cells

are identical in aggregate: both are “four copies of the multiset {1,2,3,4,5,6,7,8,9}.”

3) Remove the overlap cells (the crucial trick)

• Some cells belong to both the red set and the green set (shown as having two colors: red+green).
• For each two-colored cell:
  • Conceptually remove that cell from both the red set and the green set.

4) Analyze what remains

After removing all overlapping two-colored cells:

• The remaining red cells and remaining green cells still contain equivalent digit sets (even though you no longer know the exact digits per cell).

5) Identify the geometric consequence

• The remaining cells around the center form a ring of 16 cells surrounding the central 3×3 box.
• This ring is the Phistomefel Ring.

Core property (“mind-blowing” takeaway)

• The digits that appear in the Phistomefel ring are exactly the same digits as those appearing in the four corner 2×2 blocks of the Sudoku (matching as a multiset/collection).
• This holds for every Sudoku puzzle.
• The presenter suggests verifying it by:
  • Solving a Sudoku,
  • comparing the ring digits to the four corner 2×2 digits → they match (as a multiset).

Additional concepts mentioned

Generalization to other shapes

• The video asks whether other geometric configurations could share a similar property.
• The presenter indicates that other shapes can be constructed using the same style of equivalence reasoning.

Another example by a known constructor

• Aad van de Wetering is mentioned for finding a related trick involving:
  • a square in the top-left of a Sudoku,
  • and a square in the bottom-right of the Sudoku,
  • using the same underlying “equivalence/removal” idea.

Connection to advanced solving

• This kind of equivalence reasoning is described as useful for advanced Sudoku solving, including “fiendish puzzles.”

Related content referenced

• The video references another Numberphile/Sudoku-related video featuring mathematical sudoku.
• It also references the channel Cracking The Cryptic.

Speakers / sources featured

• Simon (main presenter; explains the proof concept)
• Brady (another participant, including questions/confirmation such as “Phistomefel ring?”)
• Aad van de Wetering (named as a Sudoku constructor who found related tricks)
• Mark (named as an “offsider” on the channel Cracking The Cryptic)
• Cracking The Cryptic (YouTube channel referenced in connection with Simon and Mark)
• Numberphile (series/brand context for the video)