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A procedure for the construction of Costas arrays of infinite size representing configurations on non-attacking queens on the chessboard is presented.

Costas arrays [

In [

This work, after reviewing the basics on Costas arrays, proceeds to present a procedural constructive proof of the existence of NAQCAs of infinite size. The existence of NAQCAs of finite size remains an open problem.

Let

Let

For Costas arrays/permutations of infinite size, the definition is slightly modified.

Let

There exist two algebraic construction techniques for Costas arrays, based on the theory of finite fields, known as the Golomb and Welch methods [

NAQCAs of infinite size exist in abundance. A procedural construction is described below, where, starting with an empty array, dots are added to its rows and columns one at a time, so that, eventually, an NAQCA is formed after countably infinitely many steps. The infinite size of the array guarantees that, at each step, any column or row that does not contain a dot yet has finitely many forbidden positions, so there are infinitely many available positions for placing the dot therein.

NAQCAs of infinite size exist.

The proof is procedural and begins with an array

For

Observe that, once a value is assigned, the procedure does not alter it any further. To check the procedure for correctness, it remains to be shown that

it produces a permutation;

it retains the nonattacking queens property;

it retains the Costas property.

This is done inductively. To begin with, initialization certainly retains both the nonattacking queens and the Costas property. Assume now that, up to a certain step, each row and column has at most one element equal to 1 and that both the nonattacking queens and Costas properties are preserved.

As each step assigns exactly one value of 1, at that point finitely many rows and columns will contain a value of 1, so that any of steps (

The new value of 1 cannot possibly be assigned to a row or a column that already contains a value of 1, as 1s get explicitly assigned by the procedure to rows/columns that contain no value of 1. So, it is always true that each row and column has at most one element equal to 1.

Any given row or column can remain without an element equal to one, namely, full of either unassigned elements or elements equal to 0, for a finite number of steps only. This is because, at each step, finitely many elements of a row or column are 0s, and, since for row

The Costas and nonattacking queens properties are preserved, by the very definitions of the steps of the procedure. Note that the Costas property constraint assigns finitely many values of 0 at the end of each step while the nonattacking queens constraint assigns values of 0 on four “rays” (the row, the column, and the two diagonals) emanating from the point where a 1 was assigned, so that every row or column of the array not containing this specific point gets at most three values of 0 assigned by this constraint on every step.

This concludes the proof.

Note that the description of the procedure in the proof above is unnecessarily restrictive. In reality, the crucial feature is that each row and column gets exactly one value of 1, so that a bijection is obtained; the two properties (Costas and nonattacking queens) are satisfied by default thanks to the definition of the steps. Therefore, the first dot can be placed anywhere, while, subsequently, steps can be applied in an arbitrary order, as long as any two consecutive applications of any particular step lie finitely many steps apart.

Note also that the term “procedure” has been used above to characterize the proposed construction technique, instead of the more common perhaps term “algorithm.” This was a deliberate decision, in order to respect the fact that an “algorithm” is usually defined as a

A procedural and constructive proof of the existence of Costas arrays of infinite size representing configurations on nonattacking queens on the chessboard was presented. The existence of such arrays in finite sizes remains an open problem: the techniques used in this work rely crucially on the infinity of the size and do not readily generalize to finite sizes.

This material is based upon works supported by the Science Foundation Ireland under Grant nos. 05/YI2/I677, 06/MI/006 (Claude Shannon Institute), and 08/RFP/MTH1164.