Shuffled equi-n-squares

TL;DR

This paper investigates properties of shuffled n-squares, showing how any set of positions can be mapped onto column transversals, and explores applications in creating complex, unpredictable sequences and transformations.

Contribution

It introduces a key result on mapping position sets in shuffled n-squares and derives bounds on shuffles needed for transformations, with novel applications to equi-n-squares.

Findings

Each set of n positions can be mapped onto a column transversal via shuffles for n <= 34 and n=37.

An upper bound of 6n + 3(-1)^{n-1} shuffles is established for transformations.

Shuffled equi-n-squares can be used to generate unpredictable sequences and simulate complex rotations.

Abstract

A formal n-square is the set of positions in an square matrix of size n. A shuffle of a formal n-square consists of independent rotations of each row and of each column. A key result turns out to be valid at least for n <= 34 and n = 37: Each set of n positions can be mapped with one shuffle onto a transversal of the columns. We consider two applications to equi-n-squares (i.e., n-matrices filled with digits 0, .., n - 1 in equal amounts). First, a shuffled equi-n-square can be seen as a torus with n colors and two orthogonal layers of n rings that can be rotated. Unlike Rubik's cube, each permutation of colored cells can be implemented with shuffles. An upper bound of $3*(-1)^{n-1} + 6n$ shuffles is derived from the key result. Our second application invokes column transversals and a process of indirection to produce theoretically unpredictable sequences of integers in shuffled equi-n-squares. Our proof of the key result involves optimizing position sets, averaging, computations based on number partitions, rotating subsets of a regular $n$-gon apart, and the use of cyclotomic polynomials. A few intermediate results need computer assistence. These efforts also generated a variety of (partially) unsolved problems. We selected eight of these for a brief discussion based on the available theoretical and computer evidence.