Question 25
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 Anton wins with infinitely many starting numbers.
Prove that Anton wins with infinitely many starting numbers.