# 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.

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### What was the gap in Ariane Papke's proof that the minimum number of sudoku clues is 17?

I was reading McGuire's paper on why the minimum number of clues in a Sudoku puzzle is 17 when I came across a curious comment: In 2008, a 17-year-old girl submitted a proof of the nonexistence of a ...
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### Is there a 9×9 Sudoku Room Square?

The following is an order 9 Room square. Copying from Wikipedia, Each cell of the array is either empty or contains an unordered pair from the set of symbols. Each symbol occurs exactly once in each ...
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### Can you efficiently determine the number of possible solutions for an arbitrary starting sudoku configuration?

I thought it would be fun to implement a solver which updates in real time to show how many possibilities remain as you fill in squares in any order. I'm able to find a number of resources explaining ...
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### Is there a Sudoku solution with a entropic line covering all cells?

I stumbled upon a problem which seems easy but is actually hard to answer. It involves the sudoku game and a commonly used custom constraint rule called "entropic lines". The rules of a ...
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### Is there an underlying graph-theory representation of Sudoku solutions?

I have been puzzling for some time about how a completed 9x9 Sudoku solution can be represented mathematically, and how that mathematical representation can be used to enumerate the different ...
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### Is the Mod2 (Modp) Reduction of an Invertible (Over The Integers) also invertible (Over $\mathbb Z_2, \mathbb Z_p)$?

Let $M:=(m_{ij})$ be a square matrix with entries in $\{ 1,2,3,.., n\}$ ,and non-zero determinant $D$ . Let $M_k$ be its reduction Mod($k$) ; $k=2,3,.., n-1$ , i.e., $M_k:=(m_{ij} \mod k)$. Is the ...
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### What is the maximum number of given numbers that still produces a minimal Sudoku? [closed]

The minimum number of given numbers that produce a valid Sudoku with a unique solution has been proven to be $17$, so it got me thinking, what is the maximum number of given numbers that still ...
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### Permutations of $9\times 9$ Sudoku square Strang 2.1.35

The question concerns a $9\times9$ sudoku square. The question is what exchanges of rows will produce another valid $9\times9$ sudoku square. I understand that rearranging the rows in each block of $3$...
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### Proving the validity-preserving nature of certain sudoku transformations using hypergraphs

I've been making a sudoku solver to get comfortable with graphs and the following "proof" popped into my head so I wanted to see if I could actually write it. Is this argument valid? Sorry ...
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### Number of sudoku puzzles vs valid chess positions

The basic question is - which one is bigger? This possibly needs some clarification: By a sudoku puzzle I mean a grid with some cells filled with numbers and others empty so that they can be filled ...
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### How Many Uniquely Enumerated $4\times4$ Sudoku Grids Exist?

I'm looking to find how many uniquely enumerated $4\times4$ Sudoku grids exist. I am aware that there are other questions with solutions to this question, however I am asking again as there seem to be ...
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### Sudoku based problem. Counting with permutations?

I have been playing Sudoku lately, and I've seen a problem with this general form: Given N items, each of which belongs alone (1-to-1) in one of N boxes, and given a list of statements of the form: &...
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### How was this Sudoku puzzle derived? Help needed to complete a possible derivation.

Solution and rules are found here. \begin{array}{ccc|ccc|ccc} 4 & 8 & 3 & 7 & 2 & 6 & 1 & 5 & 9 \\ 7 & 2 & 6 & 1 & 5 & 9 & 4 & 8 & 3\\ 1 ...
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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 ...
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### Question involving filling numbers in a grid

Here's the problem and the diagram that goes with it Fill in each empty space of the grid in the image below with a number from 1 to 8 so that every row & column contains each of these digits only ...
1 vote
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### sudoku probability

I have a completely filled in sudoku board. If the sum of all digits in each row, in each column, and in each block (the typical sudoku constraints) are all 45, what is the probability that the board ...
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### Constraint of Sudoku problem

$$\sum_{j=3q-2}^{3q}\sum_{i=3p-2}^{3p}x_{ijk}=1, \forall k = 1 :n; p,q = 1:3$$ The above constraint wants to describe what condition in the Sudoku problem? I think the constraint here is that the ...
1 vote
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### Do there exist sudoku's with clues that only contain $1$'s and $2$'s

I read this article saying it was mathematically proven that if a sudoku has a unique solution, then it has at least $17$ initial hints. In which case, is it possible to have a sudoku with $17$ ...
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### How many different "tight" sudoku puzzles are there?

How many different sudoku puzzles are there? posted a similar question and Chris Eagle's answer https://math.stackexchange.com/a/275425/26632 points to http://www.afjarvis.staff.shef.ac.uk/sudoku/...
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### Mathematical proof for minimum number of clues in sudoku

There must be at least 17 starting clues in a sudoku to be univocally solved (brute force proof) but is there any mathematical (or if you want, more elegant) proof for this or research paper that ...
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### What is the technique can be applied into this logjam in Sudoku? [closed]

The difficulty of this Sudoku is Expert. I have tried to apply swordfish, X-chain but seems like it is not valid. But I am pretty sure is my problem because I am still new to the advanced techniques. ...
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### The number of ways in which a $9$-by-$9$ grid can be filled given some conditions.

How to find the number of ways in which the above $9$-by-$9$ grid can be filled using the digits $(1-9)$ (repetition is allowed) such that all of the following conditions are satisfied: Any $3$-by-$3$...
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### Construct a compound proposition that asserts that every cell of a 9 × 9 Sudoku puzzle contains at least one number

Here is a problem from Kenneth Rosen's Discrete Mathematics and its Applications, Section 1.3 Construct a compound proposition that asserts that every cell of a 9 × 9 Sudoku puzzle contains at ...
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### Sudokus and the Distance to a Contradiction.

Consider a sudoku puzzle for which there is a unique solution. In solving the puzzle, one enters in pencil what, a priori, each of the $81$ small squares could be, given the (at least $17$) clues that ...
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### Is there a more abstract definition of a Sudoku Board

I was trying to figure out why a Sudoku board needs 17 answers at least to be defined and I was wondering if there are already ways in which you could convert a Sudoku board into a group or graph and ...
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### Maximum Sudoku pencil marks needed

Most Sudoku mobile/video games have a pencil feature which lets you record a candidate in a given cell. Some of these let you pencil in all 9 candidates, and some only a few (3 - 5). The reduction in ...
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### Sudoku prime triples

By a sudoku prime triple I mean a tripe $(p,q,r)$ of three-digit (base ten) primes which together use each of the nonzero digits $1$ to $9$ once each. I'm wondering how many such triples there are. ...
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### Counting how many Sudoku-like grids of numbers there are

Let's say we want to count $3 \times 9$ Sudoku grids, i.e. grids whose entries are taken from $\{1,\dots,9\}$ and such that no rows or $3 \times 3$ sub-grids contain repetitions. \begin{matrix} 1 &...
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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 ...
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### Number of weak-sudoku tables

We say that an $n\times n$ table of integers in $\{1,\dots,n\}$ has the weak-sudoku property if each number appears exactly once in each row and each column. The main question is: how many weak-...
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### Sudoku solution size

We know an $n$-Sudoku puzzle is with $n \times n$ subgrids consisting of $n \times n$ cells; you will fill it with numbers from $1$ to $n^2$. Candidate solution have size polynomial in $n$, and can ...
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### Ideas for modelling the rules to Sudoku using first-order logic?

I'm trying to model the rules to sudoku using first order logic. I've got the first two rules down: (The notation I use is $a_{x,y,v}$, where $v$ is the digit and $x,y$ is the coordinates of the ...
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### Figure out which region of sudoku the item is in

Suppose we have hyper sudoku. In the dark gray areas, we can have 1-9 only once. Therefore, when solving a sudoku graph, I need to figure out what region to look over, to see if any repeats occur. ...
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### 4x4 Chromatic Sudoku Graph

I was unsatisfied with existing Sudoku graphs online and my goal was to show the structure in a manner in which the regions are explicit. Nodes which share an edge cannot have be same color. Shaded ...
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### Equation to locate a square in a square

Good evening, I have been experimenting with different Sudoku checker and have come across a problem: For a nxn Sudoku where n is a square number (4,6,19,25 etcc.), there would be an n number of ...
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### How many latin-square designs are orthogonal to this 4x4 latin square design?

Where this Latin Square is similar to sudoku, in which each row has one and only one of 1,2,3 and 4, and each column has one and only one 1,2,3 and 4. Important: orthogonality means that the new ...
1 vote
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### Would an invalid Sudoku puzzle that becomes valid when you assume its validity be valid?

Suppose you have a sudoku puzzle that you will want to solve using logic. Furthermore suppose you solve the puzzle until you reach a point where a single cell can have two possible values ($a$ or $b$ ...
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### The uniqueness of sudokus after removing clues

I am creating a sudoku puzzle generator from a filled sudoku and have the following doubt. Suppose I remove one element(let it be a) from a partially filled sudoku (S) and I get multiple solutions, so ...
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### Help needed in counting ways.

In the 3rd section titled : 'Counting Solutions' for the webpage here, there is calculation as shown here that has exercise based on filling the $9$ sub-cubes of Sudoku named as $B_1, B_2, B_3$ in the ...
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### Magic Bingo Grids

In the realm of video game bingo, it is common to use magic squares to generate cards. If you have $25$ difficulty buckets for goals, then if you lay out those buckets onto a magic square, any bingo ...
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I am given two Sudoku $S_1$ and $S_2$ and I have two check whether $S_1$ can be turned into $S_2$ with the symetry operators. The two Sudoko are in a "legal" state. So a given cell (numbers $1...9$) ...