A finite goal set in the plane which is not a Winner

TL;DR

This paper presents a specific finite goal set in the plane that Player 1 cannot achieve before Player 2, challenging previous assumptions about the universality of Player 1's strategy in geometric achievement games.

Contribution

The authors construct a finite goal set with five points that cannot be constructed by Player 1 before Player 2, providing a counterexample to prior results.

Findings

Player 1 cannot always build the goal set before Player 2

The constructed goal set has 5 points

Challenges previous universal strategies in geometric games

Abstract

J. Beck has shown that if two players alternately select previously unchosen points from the plane, Player 1 can always build a congruent copy of any given finite goal set G, in spite of Player 2's efforts to stop him. We give a finite goal set G (it has 5 points) which Player 1 cannot construct before Player 2 in this achievement game played in the plane.