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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Does this linear function exist?

I'm trying to solve Sudoku using linear programming. In Sudoku, it is known that each row and column of the grid must contain number from 1 to 9 without duplicates. I need a constraint function that ...
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1 vote
2 answers
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How many different $2 \times 2$ Sudokus are there?

PROBLEM How many different $2 \times 2$ Sudokus are there? APPROACH This seems pretty easy to brute force. There are $576$ Latin squares of size $4$ (which are the sudokus without restriction on boxes)...
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2 votes
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Sudoku Puzzle with only 1 and 0 and other restrictions

For the following sudoku-style puzzle, you are given the following 9-by-9 grid, and you need to fill it in with zeros and ones satisfying the following conditions: (i) Each row, each column, and each ...
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4 votes
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Minimum Number of Clues for Unsolvable Sudoku

I am going to make a distinction between "unsolvable" and "invalid" Sudoku. A Sudoku is unsolvable if there is no way to fill in all the spaces without violating one of the rules ...
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Determinant of ModK Reduction of Matrix.

Let $M:=(m_{ij})$ be a square ($ n \times n $) 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} ...
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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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  • 507
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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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  • 831
1 vote
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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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4 votes
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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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1 answer
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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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72 votes
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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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1 vote
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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 ...
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-1 votes
1 answer
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Help by finding placeable number in $9\times 9$ Sudoku [closed]

Can someone please explain how to derive the next placeable number in the following Sudoku? I filled possible numbers as notes, but can not reduce it to one placeable number. I am stuck for hours and ...
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  • 3
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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1 vote
1 answer
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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 ...
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1 vote
1 answer
44 views

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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1 vote
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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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1 answer
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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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  • 122
1 vote
1 answer
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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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0 votes
1 answer
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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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2 votes
1 answer
256 views

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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2 votes
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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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2 votes
1 answer
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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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  • 815
8 votes
1 answer
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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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  • 1,531
1 vote
1 answer
36 views

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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1 answer
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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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2 votes
1 answer
605 views

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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4 votes
1 answer
66 views

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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0 votes
1 answer
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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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  • 439
0 votes
2 answers
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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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  • 149
0 votes
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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 ...
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1 vote
1 answer
174 views

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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0 votes
2 answers
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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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0 votes
1 answer
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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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3 votes
1 answer
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Group theory and sudoku

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 $1...9$) ...
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1 vote
1 answer
271 views

Monte Carlo to solve sudoku puzzles

I am trying to figure out how to implement a Monte Carlo method for solving sudoku. I am looking at this blog post, however I can not figure out at all where he is pulling the number 243 out of ...
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0 answers
457 views

Number Theory - Regular Magic Squares (4x4)

Ok, so I think I found a neat way to solve a $4\times4$ regular magic square (A regular square has one of each number from $0$ to $n^2-1$ and is base $n$. So here's how I did it. We have $16$ ...
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0 votes
2 answers
110 views

In how many ways can we to place an $X$ in four cells, such that there is exactly one $X$ in each row, column, and $2\times2$ outlined box?

I am a middle school student who would really appreciate it if somebody could explain how to solve this problem using simple terms. I saw this problem on this site and one other, but I am still unsure ...
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4 votes
1 answer
850 views

My Simple Combinatorial Method to Enumerate All Sudoku Solution Grids

How may possible Sudoku Solution Grids are there? The correct answer is: $6,670,903,752,021,072,936,960$ or $6,671E21$ as was proved $12$ years ago! Or was it? Their proof never really enumerated the ...
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2 votes
1 answer
256 views

How to find the rank of a sudoku without row reduction?

Furthermore, is a sudoku always of full rank? When is it full rank and when is it not? I realized these questions are quite out there and there may not be any meaningful answer to this.
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2 votes
0 answers
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Naïve approach to constructing a chaos sudoku

I was recently reminded of the jigsaw sudoku, which is a sudoku variant where each of the nine regions that need to be filled with the numbers from 1 to 9 are irregular regions instead of square ...
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0 votes
1 answer
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How to determine the size of the complete game tree for basic [M]?

You can read the rules of the game here, or actually play it free on the mobile mbrane app, but it's not required to address the question. Essentially: players take turns placing integers onto an ...
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  • 439
3 votes
0 answers
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What is the maximum number of Sudoku boards possible?

Sudoku is a $9 \times 9$ board game subjected to $4$ conditions: Only digits ($1-9$) can be used to fill a blank box. All horizontal rows to be filled with digits without repeating a digit in row. ...
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