Cyclic Squares: A New Class of Layered Combinatorial Structures

We introduce and develop the theory of cyclic squares, a new class of combinatorial objectsarising from the structured merger of cyclic Latin squares. A cyclic square of order n withm layers, denoted (n × n)m, is formed by merging m cyclic Latin squares of order n cell-wise,yielding an n × n array of ordered m-tuples in which every coordinate layer is itself a Latinsquare. We first establish the Latin square series – an exhaustive partition of all positive integersinto infinite families S(m) = {m, 2m, 4m, . . .} rooted at odd integers – and identify the cyclicseries C = {1, 2, 4, 8, 12, 16, 20, . . .} as its distinguished subfamily. We prove that the cyclic seriesconsists precisely of the orders n satisfying n ≡ 0 (mod 4) or n ∈ {1, 2}, and that this conditionis both necessary and sufficient for the merger operation to preserve the Latin property in everycoordinate. For n ≥ 8, the number of required layers is m = n/4. Explicit constructions aregiven for orders 1, 2, 4, 8, and 12, each independently verified by direct computation. Thisrevision adds a full arithmetic treatment of the grid-entry count E(n) = n2m(n): an exactidentity E(n) = n3/4 for n ≥ 8, a complete characterisation of when E(n) is a perfect square, aproof that E(n) is never a perfect cube, closed forms and constant-coefficient recurrences forE(n) and its partial sums, and a first quantitative estimate for the key-space question raisedin Open Problem 4. These results have been catalogued in the OEIS as A396280 and, for thepartial sums, in a further submission in progress. We close with a candid discussion of potential(as opposed to demonstrated) applications, and a collection of open problems.