Cannibal Animal Games: a new variant of Tic-Tac-Toe

This paper presents a new partial two-player game, called the emph{cannibal animal game}, which is a variant of Tic-Tac-Toe. The game is played on the infinite grid, where in each round a player chooses and occupies free cells. The first player Alice can occupy a cell in each turn and wins if she occupies a set of cells, the union of a subset of which is a translated, reflected and/or rotated copy of a previously agreed upon polyomino $P$ (called an emph{animal}). The objective of the second player Bob is to prevent Alice from creating her animal by occupying in each round a translated, reflected and/or rotated copy of $P$. An animal is a emph{cannibal} if Bob has a winning strategy, and a emph{non-cannibal} otherwise. This paper presents some new tools, such as the emph{bounding strategy} and the emph{punching lemma}, to classify animals into cannibals or non-cannibals. We also show that the emph{pairing strategy} works for this problem.

Coloring and Guarding Arrangements

Given an arrangement of lines in the plane, what is the minimum number $c$ of colors required to color the lines so that no cell of the arrangement is monochromatic? In this paper we give bounds on the number c both for the above question, as well as some of its variations. We redefine these problems as geometric hypergraph coloring problems. If we define $Hlinecell$ as the hypergraph where vertices are lines and edges represent cells of the arrangement, the answer to the above question is equal to the chromatic number of this hypergraph. We prove that this chromatic number is between $Omega (log n / loglog n)$. and $O(sqrt{n})$. Similarly, we give bounds on the minimum size of a subset $S$ of the intersections of the lines in $mathcal{A}$ such that every cell is bounded by at least one of the vertices in $S$. This may be seen as a problem on guarding cells with vertices when the lines act as obstacles. The problem can also be defined as the minimum vertex cover problem in the hypergraph $Hvertexcell$, the vertices of which are the line intersections, and the hyperedges are vertices of a cell. Analogously, we consider the problem of touching the lines with a minimum subset of the cells of the arrangement, which we identify as the minimum vertex cover problem in the $Hcellzone$ hypergraph.

Blocking Coloured Point Sets

This paper studies problems related to visibility among points in the plane. A point $x$ emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $bar{vw}$. A set of points $P$ is emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,win P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets ${n_1,n_2,n_3,n_4}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points.

Cauchys Arm Lemma on a Growing Sphere

We propose a variant of Cauchys Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogovs Comparison Theorem, Legendres Theorem and Cauchys Arm Lemma.