Anton chooses as starting number an integer $n \ge 0$ which is not a square. Berta adds to this number its successor $n+1$. If this sum is a perfect square, she has won. Otherwise, Anton adds to this sum, the subsequent number $n+2$. If this sum is a perfect square, he has won. Otherwise, it is again Berta's turn and she adds the subsequent number $n+3$, and so on. Prove that there are infinitely many starting numbers, leading to Anton's win. (Richard Henner)