The Anti-Knight Killer Sudoku - Numberphile

Key Concepts

• Fistel Ring: A pattern in Sudoku puzzles where the 16 digits surrounding the central 3x3 box are identical to the digits in the four 2x2 corner boxes.
• Knight's Move Constraint: A Sudoku variant rule where cells a knight's move apart in chess cannot contain the same digit.
• Anti-Killer Sudoku: A Sudoku variant that combines standard Sudoku rules with Killer Sudoku cages (groups of cells with a specified sum) and an additional constraint (in this case, the Knight's Move constraint).
• Potato Head 21: The constructor of the "Anti-Killer Sudoku" puzzle discussed.
• Fistel Ring Property with Knight's Move Constraint: In a Sudoku with a Knight's Move constraint, the Fistel Ring must contain either all nine distinct Sudoku digits or exactly eight of them. It cannot miss two or more digits.

The Fistel Ring and its Extension in Anti-Killer Sudoku

The discussion begins by revisiting the "Fistel Ring," a pattern previously identified in all Sudoku puzzles. This pattern states that the 16 digits surrounding the central 3x3 box are identical to the digits found in the four 2x2 corner boxes. This trait was discovered by a German constructor named Fistel.

The "Anti-Killer Sudoku" Puzzle

The main focus of the video is a specific Sudoku puzzle created by "Potato Head 21" called "Anti-Killer Sudoku." This puzzle adheres to standard Sudoku rules (digits 1-9 in each row, column, and 3x3 box) and includes Killer Sudoku cages with specified sums (e.g., four digits summing to 15).

A crucial additional rule is the Knight's Move Constraint: cells a knight's move apart in chess cannot contain the same digit. For instance, if the central cell contains a '1', then any cell a knight's move away from it cannot be a '1'.

A notable characteristic of this puzzle is its lack of initial given digits, a trend becoming more common in variant Sudoku.

Solving the Puzzle: The Initial Challenge and the Fistel Ring Insight

The presenter recounts their initial struggle to solve this puzzle, taking 33 minutes to place just one digit. Upon revisiting it, the same digit was placed in 1 minute, highlighting the importance of understanding a specific underlying principle.

The key to solving this puzzle lies in the interaction between the Fistel Ring and the Knight's Move constraint. The presenter reveals Potato Head's discovery: in a Sudoku with a Knight's Move constraint, the Fistel Ring must contain either all nine distinct Sudoku digits or eight of them. It cannot miss two or more digits.

Proof of the Fistel Ring Property with Knight's Move Constraint

The presenter explains the logic behind this property:

1. Fistel Ring Equivalence: The previous video established that the digits in the red cells (the ring around the center) are the same as the digits in the blue cells (the 2x2 corners).
2. Hypothetical Missing Digit: Imagine a digit (e.g., '7') is not present in the Fistel Ring (red cells). Due to the equivalence, it's also not in the corner 2x2 boxes (blue cells).
3. Domino Placement: In each of the four corner boxes, this missing digit would have to reside within specific pairs of cells that form "dominoes" (two adjacent cells).
4. Knight's Move Interaction: Now, consider the Knight's Move constraint. If a '7' were placed in one of the cells of a domino in a corner box, the Knight's Move constraint would prevent it from being placed in certain other cells within that same box and potentially in adjacent boxes.
5. Central Cell Necessity: The presenter demonstrates that if a digit is missing from the Fistel Ring, the Knight's Move constraint, combined with the standard Sudoku rules, forces that digit to be placed in the absolute center cell of the 9x9 grid.
6. Uniqueness of Missing Digit: If a second digit were also missing from the Fistel Ring, it would also be forced into the center cell, which is impossible as only one digit can occupy a cell. Therefore, at most, only one digit can be missing from the Fistel Ring in a Sudoku with a Knight's Move constraint.

Applying the Principle to the "Anti-Killer Sudoku"

The presenter then applies this knowledge to the "Anti-Killer Sudoku":

• Cage Sums and Digit Exclusion: The sums of the Killer Sudoku cages are crucial. For example, the cage summing to 11 with four distinct digits can only be formed by 1, 2, 3, and 5. This implies that higher digits (like 7, 8, 9) are difficult to fit into these cages.
• Identifying the Missing Digit: By analyzing the possible combinations for the cage sums, particularly the cage summing to 15, it's determined that it must contain either an 8 or a 9. The cage summing to 4, when combined with the constraints, forces the inclusion of 1, 2, and 3. This leads to the conclusion that the digit '9' must be the one missing from the Fistel Ring, and therefore, it must be in the center cell.

Broader Applicability of the Principle

The presenter notes that this principle is not exclusive to Knight's Move constraints. It also applies to Sudoku puzzles where the two main diagonals are restricted from having repeated digits. In such cases, the "naughty digit" (the one missing from the relevant ring) is also forced into the central cell.

The Impact of Knowing the Theory

Knowing that the Fistel Ring could be missing a digit (and which one) significantly simplifies the puzzle. Once the '9' is placed in the center, the puzzle becomes much more manageable, akin to a standard Sudoku.

Further Puzzle Design and Conclusion

The discussion touches upon other puzzles designed by Potato Head 21 that leverage this Fistel Ring property, such as a puzzle where the Fistel Ring must sum to 109. This sum implies a specific set of digits must be present to maximize the sum while still adhering to the rule of having at least eight distinct digits.

The presenter expresses admiration for the ingenuity of puzzle constructors like Potato Head 21, emphasizing the intellectual stimulation these puzzles provide. The video concludes with a promotion for Brilliant.org, an online learning platform offering courses in math, computer science, and other subjects, highlighting its interactive nature and offering a discount for Numberphile viewers.

The final segment of the transcript includes a brief, lighthearted exchange about breakfast habits, which is unrelated to the Sudoku discussion.