Questions tagged [sudoku]

Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a $9\times9$ grid with digits so that each column, each row, and each of the nine $3\times3$ subgrids that compose the grid (also called "boxes", "blocks", "regions" or "subsquares") contain all the digits from $1$ to $9$. The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a unique solution.

Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a $9\times9$ grid with digits so that each row, column and the nine $3\times3$ subgrids that compose the grid (also called "boxes", "blocks", "regions" or "subsquares") contain all the digits from $1$ to $9$. The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a unique solution.

A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any of the nine blocks (or boxes of $3\times 3$ cells). The relationship between the two theories is known, after it was proven that a first-order formula that does not mention blocks is valid for Sudoku if and only if it is valid for Latin squares.

The general problem of solving Sudoku puzzles on $n^2\times n^2$ grids of $n\times n$ blocks is known to be NP-complete

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Are there any valid continuous Sudoku grids?

A standard Sudoku is a $9\times 9$ grid filled with digits such that every row, column, and $3\times 3$ box contains all the integers from $1$ to $9$. I am thinking about a generalization of Sudoku which I call "continuous Sudoku", which consists of…
ZKG
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Is a Sudoku a Cayley table for a group?

I want to know if the popular Sudoku puzzle is a Cayley table for a group. Methods I've looked at: Someone I've spoken to told me they're not because counting the number of puzzle solutions against the number of tables with certain permutations of…
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Sudokus as composition tables of finite groups

If $G$ is a finite group then the composition table of $G$ is a latin square (ie, each row and column contains each group element exactly once). Assume now that $|G| = n^2$ for some natural number $n$. We can then split the composition table for $G$…
Tobias Kildetoft
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Sudoku with special properties

Sudoku is a puzzle, with the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid (also "sudoku-blocks") contains all of the digits from 1 to 9. Let's define block as a…
rfg
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Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but this does not seem to help establishing …
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Maximum number of clues in a Sudoku game that does not produce a unique solution

You may have heard that recently it was proven that the smallest number of starting clues for a Sudoku game, guaranteeing a unique solution, is 17. An example is shown below. I am interested in the opposite: What is the largest number of starting…
Tom
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Can a solved Sudoku game have an invalid region if all rows and columns are valid?

Given a $9 \times 9$ solved Sudoku game with $3 \times 3$ regions, is it possible that one (or more) of the regions are invalid if all rows and columns are valid (i.e. have a unique sequence of $1-9$)?
dragonfly
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The Hardest Sudoku Puzzle

I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?" My response: "A Sudoku puzzle with the minimal amount of starting numbers where the puzzle is still solvable." However, I am…
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Determining if two Sudoku boards are in the same equivalence class

Consider the following $9 \times 9$ Sudoku board 963 174 258 178 325 649 254 689 731 821 437 596 496 852 317 735 961 824 589 713 462 317 246 985 642 598 173 and the following partially filled Sudoku board .6. 1.4 .5. 2.. ...…
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Is there a sudoku (Latin Square Pattern) state in a Rubik's cube $6\times6\times6?$

Suppose, Initial state Rubik's Cube 6x6x6 444444 444444 444444 444444 444444 444444 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222…
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How many Sudoku puzzles are there with at least one solution?

A Sudoku puzzle is a 9 by 9 matrix of blanks(which we can represent as 0), and elements of the set {1,2,3,4,5,6,7,8,9}. How many Sudoku puzzles are there with at least one solution. Yes, I am even counting grids with no blanks.
user107952
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Sudoku: Maximal minimum number of starting clues

It is well known (as shown here) that the minimum number of starting clues a Sudoku puzzle may have to generate a unique solution is 17. My main question is Given a completed Sudoku grid, is it always possible to find a subset of 17 starting clues…
hexomino
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Diagonal-free Sudoku grid

I have a Sudoku grid with the property that diagonally adjacent elements are distinct (it is also a torus under the same property). My question is up to isomorphism, is the grid unique? Here's the grid: $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline 6&…
JMP
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Calculating Gröbner basis for Sudoku

I'm trying to write a program that solves sudokus using a Gröbner basis. I introduced 81 variables $x_1$ to $x_{81}$, this is a linearisation of the sudoku board. The space of valid sudokus is defined by: for $i=1,\ldots,81$ : $F_i = (x_i - 1)(x_i -…
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Sudoku puzzles and propositional logic

I am currently reading about how to solve Sudoku puzzles using propositional logic. More specifically, they use the compound statement $$\bigwedge_{i=1}^{9} \bigwedge_{n=1}^{9} \bigvee_{j=1}^{9}~p(i,j,n)$$ where $p(i,j,n)$ is the proposition that…
Mack
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