Question 54
Let $n$ be a given even positive integer. First John writes the consecutive square numbers $1^2, 3^2, \dots, (2n-1)^2$ on the blackboard. Then he picks some three of them, say $a_1, a_2, a_3$, erases them and writes the number
$$
1 + \sum_{1 \le i \lt j \le 3} |a_i - a_j|
$$
on the blackboard. He continues this way replacing each time three of the numbers on the blackboard until only two numbers remain on it. Prove that the sum of the squares of the two remaining numbers is different from any of the numbers $1^2, 3^2, \dots, (2n-1)^2$ written initially on the blackboard.
$$
1 + \sum_{1 \le i \lt j \le 3} |a_i - a_j|
$$
on the blackboard. He continues this way replacing each time three of the numbers on the blackboard until only two numbers remain on it. Prove that the sum of the squares of the two remaining numbers is different from any of the numbers $1^2, 3^2, \dots, (2n-1)^2$ written initially on the blackboard.