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

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.