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How many chess rooks or queens does it take to guard all squares of a given polyomino? This question is a version of the art gallery problem in which the guards can ‘’see’’ whichever squares the rook or queen attacks. We show that n/2 rooks or n/3 queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes (this is joint work with Hannah Alpert).
Erika Roldán is currently a Marie Skłodowska-Curie Fellow in the EuroTechPostdoc Programme at the Technische Universität München (TUM) and EPFL Lausanne. Her research interests include biomathematics, stochastic topology, topological and geometric data analysis, extremal topological combinatorics, discrete configuration spaces, recreational mathematics, learning analytics, and educational technology.
We will be discussing the topic of John Conway’s Rational Tangles, inspired by a Math Circle activity described by Tom Davis. Participants should aim to have read through Tom’s writeup of the activity, and discussion can focus on either the topic itself or ways to run activities on it, depending on the preferences of those who attend.
The paper can also be found in this folder along with some more advanced papers for those who wish to read further, although this is not required in order to attend.