Twin Sudoku Puzzles and Triplet Solid Sudoku Cubes From Strongly Mutually Distinct Twin Sudoku Tables

Abstract

A new class of twin <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sudoku</i> tables (TSTs) is presented. These tables can be divided into both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s \times d$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d \times s$</tex-math></inline-formula> subtables. They are constructed using the cyclotomic cosets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Z_n$</tex-math></inline-formula> via two distinct vectors of cyclotomic coset elements and their Kronecker product. We prove that it is possible to generate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> TSTs that are strongly mutually distinct (SMD), i.e., for every <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0\leq i, j \leq m-1$</tex-math></inline-formula> , the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(i,j)$</tex-math></inline-formula> th entry of the tables contains different symbols. We also provide a method to construct <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> different TSTs that can be converted into twin solid <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sudoku</i> tables (TSSTs) as a perfect set of SMD TSSTs in order to make triplet solid <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sudoku</i> cubes (TSSCs). These TSSCs are symmetric cubes so that a cut from any of the six faces is a TSST. As a result, new twin <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sudoku</i> puzzles (TSPs) and SMDTSPs are obtained that can be used to design new types of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sudoku</i> games.