Question 14
Initially on the board, all the integers from $1$ to $2002$ inclusive, without repetitions, are written in one line and in some order.
In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step.
After performing $2001$ steps, only one number remains on the board.
Determine all possible values of the number left on the board by varying the order of the $2002$ numbers on the initial line (and performing the $2001$ steps).
In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step.
After performing $2001$ steps, only one number remains on the board.
Determine all possible values of the number left on the board by varying the order of the $2002$ numbers on the initial line (and performing the $2001$ steps).