Beyond Sudoku: New puzzle variants based on a novel mathematical mapping

This paper introduces a novel class of Latin squares that includes Sudoku squares as a subset. We define a new structure by applying a ceiling-based function to each entry of a Latin square of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> , grouping values into blocks, and requiring that each block contains each integer from 1 to r exactly r times. This defines an intermediate grid type that bridges standard Latin squares and Sudoku grids. Furthermore, we introduce a bijective mapping ϕ on the space of Latin squares, where each row in ϕA encodes the column positions of a given value in the original square A . This mapping enables a new method for generating Sudoku-like grids by transforming Latin squares in a structured way. We present mathematical foundations, formal definitions, and proofs establishing the properties of this new class of grids and their relation to classical Sudoku puzzles.