lattice point game
Source: Austrian - Polish 1999 APMC
May 4, 2020
latticegamegame strategycombinatoricscombinatorial geometry
Problem Statement
A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold:
(i) The endpoints of each selected segment are lattice points;
(ii) Each selected segment is parallel to a coordinate axis or to one of the lines ,
(iii) Each selected segment contains exactly five lattice points, all of which are selected,
(iv) Every two selected segments have at most one common point.
A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.