# A Sudoku Secret to Blow Your Mind - Numberphile

TLDRThe video discusses the Phistomefel Ring, a fascinating property of Sudoku puzzles discovered by a German constructor. It involves using set equivalence theory to identify a ring of 16 cells around the central 3x3 box, which contains the same digits as the four corner 2x2 squares. This property can help solve certain Sudoku puzzles and has led to the discovery of other geometric configurations with similar properties, showcasing the mathematical beauty within Sudoku.

###### Takeaways

- π§© The Phistomefel Ring is a property in Sudoku puzzles, named after a German constructor who discovered it.
- π’ It's a lesser-known property that exists in all Sudoku puzzles and has a fascinating proof.
- π Sudoku rules involve placing the numbers 1 to 9 once each in every row, column, and 3x3 box.
- π΄ The concept of 'set equivalence theory' is introduced, highlighting how certain groups of cells contain complete sets of numbers 1 to 9.
- π’ By focusing on cells with two colors (red and green), one can understand the Phistomefel Ring's significance.
- π The Phistomefel Ring consists of 16 cells ringing the central 3x3 box of the Sudoku, which share the same digits as the four corner 2x2 boxes.
- π© This property can be applied to any Sudoku puzzle, offering a unique way to solve or verify solutions.
- π Other shapes and configurations on a Sudoku grid can also exhibit similar properties, as discovered by Dutch constructor Aad van de Wetering.
- π The equivalence of different parts of the grid's geometry can be crucial for solving advanced Sudoku puzzles.
- π The mathematical relationships within Sudoku can lead to beautiful mathematical results and equations.
- πΊ For further exploration, there are resources available, including videos and channels dedicated to mathematical Sudoku and puzzle-solving.

###### Q & A

### What is the Phistomefel Ring in Sudoku?

-The Phistomefel Ring is a property of Sudoku puzzles discovered by a German Sudoku constructor named Phistomefel. It is a set of cells that, despite not having their individual digits known, collectively contain the same set of digits 1 to 9 as the four corners of the Sudoku grid.

### Who discovered the Phistomefel Ring?

-The Phistomefel Ring was discovered by a German Sudoku constructor named Phistomefel.

### How does the Phistomefel Ring help in solving Sudoku puzzles?

-The Phistomefel Ring can assist in solving Sudoku puzzles by providing a deeper understanding of the relationships between different parts of the grid. It helps to establish equivalences between sets of cells that contain the same set of digits 1 to 9, which can be useful in more advanced solving techniques.

### What is Set Equivalence Theory in the context of Sudoku?

-Set Equivalence Theory in Sudoku is a technique that involves identifying sets of cells that contain the same exact set of digits 1 to 9. This theory is used to establish relationships between different parts of the grid, which can help in solving the puzzle.

### How many sets of digits 1 to 9 are there in a standard Sudoku puzzle?

-In a standard Sudoku puzzle, there are several sets of digits 1 to 9: each row, each column, and each of the nine 3x3 grids contain a unique set of digits from 1 to 9.

###### Outlines

##### π§© Introducing the Phistomefel Ring in Sudoku

This paragraph introduces the concept of the Phistomefel Ring, a fascinating property in sudoku puzzles discovered by a German constructor named Phistomefel. It explains that this property is present in all sudoku puzzles but is not widely known. The speaker discusses the Phistomefel Ring's potential to aid in solving certain sudoku puzzles and provides an example using set equivalence theory. The explanation includes highlighting specific cells in a sudoku grid, demonstrating how the red and green sets of cells contain the same sets of digits 1 to 9. The concept is further illustrated by focusing on a cell that is part of both sets and hypothesizing its removal to show that the remaining cells in both sets remain equivalent. The segment concludes with the identification of the Phistomefel Ring as a ring of 16 cells around the central 3x3 box of the sudoku, which shares the same digits as the four corner 2x2 squares. The speaker also mentions the possibility of discovering other shapes or configurations with similar properties, referencing the work of Dutch constructor Aad van de Wetering.

##### π Advanced Sudoku Solving and Mathematical Insights

The second paragraph delves into the significance of understanding different parts of a sudoku grid's geometry for advanced puzzle-solving. It mentions that the equivalence of various grid sections can be crucial in cracking complex sudoku puzzles. The speaker encourages viewers to explore further by checking out links to another video on mathematical sudoku and a YouTube channel called 'Cracking The Cryptic' for more content. The paragraph concludes with a thank you note to Patreon supporters, mentioning a unique way of signing and mailing out prime numbers to them, with a playful hint at sending twin primes to some.

###### Mindmap

###### Keywords

##### π‘Sudoku

##### π‘Phistomefel Ring

##### π‘Set Equivalence Theory

##### π‘Numberphile

##### π‘Aad van de Wetering

##### π‘Difference of Squares Equation

##### π‘Cracking The Cryptic

##### π‘Patreon Supporters

##### π‘Unique Primes

##### π‘Twin Primes

###### Highlights

The discussion revolves around the concept of the Phistomefel Ring in Sudoku, a lesser-known property that can aid in solving puzzles.

The Phistomefel Ring is named after a German Sudoku constructor, Phistomefel, who discovered it a few years ago.

The property of the Phistomefel Ring exists in all Sudoku puzzles, yet it is not widely known.

The proof of the Phistomefel Ring is described as quite lovely and potentially mind-blowing.

Set Equivalence Theory is introduced as a method to understand the Phistomefel Ring.

The demonstration involves highlighting certain cells in red and green, representing different sets of numbers from the Sudoku grid.

The red cells contain four complete sets of the digits 1 to 9, which is a crucial aspect of the Phistomefel Ring.

The green cells, like the red cells, also represent four sets of the digits 1 to 9, establishing an equivalence between the two sets.

The Phistomefel Ring is identified as the 16 cells ringing the central box of the Sudoku, which have the same digits as the four corners of the puzzle.

The Phistomefel Ring's property is applicable to every Sudoku puzzle, making it a universal feature.

The concept can be extended to create different shapes or configurations on a Sudoku grid that possess similar properties.

A Dutch constructor named Aad van de Wetering discovered another trick related to the Phistomefel Ring, involving squares in opposite corners of the Sudoku.

The geometry of Sudoku can yield beautiful mathematical results when applying the equivalence of different parts of the grid.

Advanced Sudoku solving often relies on understanding the equivalence of various grid parts, which can help crack complex puzzles.

The video encourages viewers to try the Phistomefel Ring on any Sudoku and compare the central box digits to the corners' 2x2s to verify its effectiveness.

The video concludes by directing viewers to additional resources for learning more about mathematical Sudoku and advanced solving techniques.

###### Transcripts

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