#### Vol. 9, No. 4, 2016

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Other MSP Journals
Arranging kings $k$-dependently on hexagonal chessboards

### Robert Doughty, Jessica Gonda, Adriana Morales, Berkeley Reiswig, Josiah Reiswig, Katherine Slyman and Daniel Pritikin

Vol. 9 (2016), No. 4, 699–713
##### Abstract

Tessellate the plane into rows of hexagons. Consider a subset of $2n$ rows of these hexagons, each row containing $2n$ hexagons, forming a rhombus-shaped chessboard of $4{n}^{2}$ spaces. Two kings placed on the board are said to “attack” each other if their spaces share a side or corner. Placing kings in alternating spaces of every other row results in an arrangement where no two of the ${n}^{2}$ kings are attacking each other. According to our specific distance metric, ${n}^{2}$ is in fact the largest number of kings that can be placed on such a board with no two kings attacking one another, for a maximum “density” of $\frac{1}{4}$. We consider a generalization of this maximum density problem, instead requiring that no king attacks more than $k$ other kings for $0\le k\le 12$. For instance when $k=2$ the density is at most $\frac{1}{3}$. For each $k$ we give constructive lower bounds on the density, and use systems of inequalities and discharging arguments to yield upper bounds, where the bounds match in most cases.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/involve

We have not been able to recognize your IP address 52.87.176.39 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.

##### Keywords
$k$-dependence, combinatorial chessboard, optimization, discharging, linear programming
##### Mathematical Subject Classification 2010
Primary: 90C05, 90C27