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Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner's minimum no-3-in-a-line problem

Seunghwan Oh, John R. Schmitt, Xianzhi Wang

Abstract

In 1976 Martin Gardner posed the following problem: ``What is the smallest number of [queens] you can put on an [$n \times n$ chessboard] such that no [queen] can be added without creating three in a row, a column, or a diagonal?'' The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least $n$, except in the case when $n$ is congruent to $3$ modulo $4$, in which case one less may suffice. When $n>1$ is odd, Gardner conjectured the lower bound to be $n+1$. We prove this conjecture in the case that $n$ is congruent to 1 modulo 4. The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica.

Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner's minimum no-3-in-a-line problem

Abstract

In 1976 Martin Gardner posed the following problem: ``What is the smallest number of [queens] you can put on an [ chessboard] such that no [queen] can be added without creating three in a row, a column, or a diagonal?'' The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least , except in the case when is congruent to modulo , in which case one less may suffice. When is odd, Gardner conjectured the lower bound to be . We prove this conjecture in the case that is congruent to 1 modulo 4. The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica.
Paper Structure (7 sections, 4 theorems, 29 equations, 4 figures, 1 table)

This paper contains 7 sections, 4 theorems, 29 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction.
  2. Definitions and notation
  3. Proof of Main Theorem
  4. Concluding remarks
  5. Placements that correspond to the zero vector
  6. Other cases to consider
  7. Acknowledgments

Key Result

Theorem 1

[Cooper, Pikhurko, Schmitt, Warrington cooperpikhurkoschmittwarrington:14] For $n \geq 1$, the answer to Gardner's no-3-in-a-line queens version problem is at least $n$, except in the case when $n$ is congruent to $3$ modulo $4$, in which case one less may suffice.

Figures (4)

  • Figure 1: An example of a maximal placement with $10$ queens on a $9 \times9$ board.
  • Figure 2: An example of a good placement with one lonely queen on a $5 \times 5$ board.
  • Figure 3: A placement of 10 queens on a $9 \times 9$ board which corresponds to the zero vector in Nul $A$. Sqaures marked with a cross indicate those squares for which the additional placement of a queen would not yield three-in-a-line.
  • Figure 4: The $8k$ squares that form the perimeter of $U$ and some covering lines.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4