A Sudoku Secret to Blow Your Mind - Numberphile

Numberphile

4 Jan 202406:08

EducationalLearning

32 Likes 10 Comments

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

🧩 The Phistomefel Ring is a property in Sudoku puzzles, named after a German constructor who discovered it.
🔢 It's a lesser-known property that exists in all Sudoku puzzles and has a fascinating proof.
📏 Sudoku rules involve placing the numbers 1 to 9 once each in every row, column, and 3x3 box.
🔴 The concept of 'set equivalence theory' is introduced, highlighting how certain groups of cells contain complete sets of numbers 1 to 9.
🟢 By focusing on cells with two colors (red and green), one can understand the Phistomefel Ring's significance.
🔄 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.
🎩 This property can be applied to any Sudoku puzzle, offering a unique way to solve or verify solutions.
🔍 Other shapes and configurations on a Sudoku grid can also exhibit similar properties, as discovered by Dutch constructor Aad van de Wetering.
🔑 The equivalence of different parts of the grid's geometry can be crucial for solving advanced Sudoku puzzles.
📈 The mathematical relationships within Sudoku can lead to beautiful mathematical results and equations.
📺 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

00:00

🧩 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.

05:07

🚀 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

💡Sudoku

Sudoku is a logic-based, combinatorial number-placement puzzle where the objective is to fill a 9x9 grid with digits so that each column, each row, and each of the nine 3x3 subgrids that compose the grid (also called 'boxes') contain all of the digits from 1 to 9. It is a popular puzzle game that requires patience and logical thinking to solve.

💡Phistomefel Ring

The Phistomefel Ring is a unique property of Sudoku puzzles discovered by a German Sudoku constructor named Phistomefel. It refers to a specific arrangement of cells in a Sudoku grid that, despite the variability of the puzzle's solution, always contains the same set of digits as the four 2x2 squares in the corners of the grid. This property can be used to solve certain Sudoku puzzles and adds an extra layer of intrigue to the game.

💡Set Equivalence Theory

Set Equivalence Theory, as discussed in the context of Sudoku, is a method of solving puzzles by identifying sets of cells that must contain the same digits. This theory is based on the principle that if two groups of cells each contain a complete set of the digits 1 to 9, and a cell is shared between these groups, then removing that cell from both groups does not change the fact that the remaining cells in each group still contain the same set of digits.

💡Numberphile

Numberphile is a popular YouTube channel that focuses on the world of numbers and mathematics, often exploring interesting mathematical concepts, puzzles, and the beauty of numbers. The channel features various presenters discussing a wide range of topics related to numbers, including Sudoku and the Phistomefel Ring.

💡Aad van de Wetering

Aad van de Wetering is a renowned Dutch Sudoku constructor who has contributed to the field by discovering additional properties and tricks related to Sudoku puzzles. His work has expanded the understanding of the mathematical relationships within Sudoku, leading to the creation of more complex and interesting puzzles.

💡Difference of Squares Equation

The Difference of Squares Equation is a mathematical formula that represents the difference between two squares of numbers as a product of two binomials. In the context of Sudoku, this equation can be applied to certain configurations within the grid to solve puzzles or reveal hidden properties.

💡Cracking The Cryptic

Cracking The Cryptic is a YouTube channel that focuses on solving a variety of puzzles, including Sudoku and other logic-based games. The channel's hosts, including Simon, explore different types of puzzles and share their strategies and insights for solving them, often delving into the mathematical principles behind the puzzles.

💡Patreon Supporters

Patreon Supporters refers to individuals who financially support content creators on the Patreon platform. These supporters often receive exclusive content, updates, or other benefits as a thank you for their contributions. In the context of the video, Patreon supporters are acknowledged for their role in enabling the production of content.

💡Unique Primes

Unique Primes refers to prime numbers that have not been used before and are being sent as a special thank you to Patreon supporters in the video's narrative. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

💡Twin Primes

Twin primes are pairs of prime numbers that differ by two, such as (3, 5) or (11, 13). They are of interest in number theory and have been the subject of various conjectures and studies. In the context of the video, twin primes are mentioned as a special type of prime number that might be sent to some Patreon supporters.

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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