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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
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Abstract |
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Tessellate the plane into rows of hexagons. Consider a subset of rows of these hexagons, each row containing hexagons, forming a rhombus-shaped chessboard of 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 kings are attacking each other. According to our specific distance metric, 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 . We consider a generalization of this maximum density problem, instead requiring that no king attacks more than other kings for . For instance when the density is at most . For each 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. |
Keywords
$k$-dependence, combinatorial chessboard, optimization,
discharging, linear programming
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Mathematical Subject Classification 2010
Primary: 90C05, 90C27
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Milestones
Received: 22 July 2015
Revised: 31 July 2015
Accepted: 17 September 2015
Published: 6 July 2016
Communicated by Arthur T. Benjamin
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Authors |
| Robert Doughty | |
| Miami University Oxford, OH 45162 United States |
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| Jessica Gonda | |
| The University of Akron Akron, OH 44304 United States |
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| Adriana Morales | |
| University of Puerto Rico San Juan, 00931 Puerto Rico |
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| Berkeley Reiswig | |
| Anderson University Anderson, SC 29621 United States |
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| Josiah Reiswig | |
| University of South Carolina Columbia, SC 29208 United States |
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| Katherine Slyman | |
| Wake Forest University Winston-Salem, NC 27106 United States |
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| Daniel Pritikin | |
| Department of Mathematics Miami University 123 Bachelor Hall 301 S. Patterson Ave. Oxford, OH 45056 United States |
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