Sudoku Rectangle Completion

TL;DR

This paper explores the extension of Latin square completion results to Sudoku squares and derives asymptotic bounds on their quantity, enhancing understanding of Sudoku's combinatorial properties.

Contribution

It extends classical Latin square completion results to Sudoku squares and develops a procedure to estimate their asymptotic counts.

Findings

Extended Latin square completion results to Sudoku squares.

Derived asymptotic bounds on the number of Sudoku squares.

Provided a new method for analyzing Sudoku square enumeration.

Abstract

Over the last decade, Sudoku, a combinatorial number-placement puzzle, has become a favorite pastimes of many all around the world. In this puzzle, the task is to complete a partially filled $9 \times 9$ square with numbers 1 through 9, subject to the constraint that each number must appear once in each row, each column, and each of the nine $3 \times 3$ blocks. Sudoku squares can be considered a subclass of the well-studied class of Latin squares. In this paper, we study natural extensions of a classical result on Latin square completion to Sudoku squares. Furthermore, we use the procedure developed in the proof to obtain asymptotic bounds on the number of Sudoku squares of order $n$.