Related papers: Latin Puzzles
In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for…
Sudoku grids can be thought of as graphs where the vertices are the squares of the grid, and edges join vertices in the same row, column, or sub-grid. A Sudoku puzzle corresponds to a partial proper coloring of the Sudoku graph. We provide…
The partial Latin square extension problem is to fill as many as possible empty cells of a partially filled Latin square. This problem is a useful model for a wide range of applications in diverse domains. This paper presents the first…
To get another from a given latin square, we have to change at least 4 entries. We show how to find these entries and how to change them.
We provide several algorithms for the exact, uniform random sampling of Latin squares and Sudoku matrices via probabilistic divide-and-conquer (PDC). Our approach divides the sample space into smaller pieces, samples each separately, and…
Sudoku is a popular combinatorial puzzle. A new method of solving Sudoku is presented, which involves formulating a puzzle as a special type of transportation problem. This model allows one to solve puzzles with more than one solution,…
We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules. We focus on the standard puzzles for which the enumeration problems can be solved by explicit formulas or by classical numbers,…
In this paper we try to answer the question "What constitutes Sudoku difficulty rating across different Sudoku websites?" Using two distinct methods that can both solve every Sudoku puzzle, I propose two new metrics to characterize Sudoku…
We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$…
How can we predict the difficulty of a Sudoku puzzle? We give an overview of difficulty rating metrics and evaluate them on extensive dataset on human problem solving (more then 1700 Sudoku puzzles, hundreds of solvers). The best results…
A partial Latin square (PLS) is a partial assignment of n symbols to an nxn grid such that, in each row and in each column, each symbol appears at most once. The partial Latin square extension problem is an NP-hard problem that asks for a…
An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely…
A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares…
In this study, we investigate the computational complexity of some variants of generalized puzzles. We are provided with two sets S_1 and S_2 of polyominoes. The first puzzle asks us to form the same shape using polyominoes in S_1 and S_2.…
A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two adjacent cells (containing integers) to be their difference modulo $n$,…
This article describes how to solve Sudoku puzzles using Quadratic Unconstrained Binary Optimization (QUBO). To this end, a QUBO instance with 729 variables is constructed, encoding a Sudoku grid with all constraints in place, which is then…
A Latin square $L(n,k)$ is a square of order $n$ with its entries colored with $k$ colors so that all the entries in a row or column have different colors. Let $d(L(n,k))$ be the minimal number of colored entries of an $n \times n$ square…
Quantum magic squares were recently introduced as a 'magical' combination of quantum measurements. In contrast to quantum measurements, they cannot be purified (i.e. dilated to a quantum permutation matrix) -- only the so-called…
A classical question in combinatorics is the following: given a partial latin square P, when can we complete P to a latin square L? In this paper, we will investigate the class of \leq\epsilon-dense partial latin squares: partial latin…
We generalize classical triangular Schubert puzzles to puzzles with convex polygonal boundary. We give these puzzles a geometric Schubert calculus interpretation and derive novel combinatorial commutativity statements, using purely…