The numbers $1, 2, …,1000$ are written on a blackboard, in some order. Between every pair of consecutive terms, the absolute difference of the two terms is written between them, and then all the original numbers are erased. (In other words, if the numbers are $a_1, a_2, …, a_n$, they are replaced by the numbers $|a_1−a_2|, |a_2−a_3|, …, |a_{(n−1)}−a_n|$.) This procedure is repeated $999$ times, when there is only one number left on the blackboard. What is the largest possible value of this last number?
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The sequence $\{1000, 1,2,\ldots,999\}$ yields a largest possible value of $998$.
The first subtraction must reduce the maximum value by some fiigure, and does not yield any zeros, therefore the second subtraction must also reduce the maximum number. Therefore the minimum reduction is $2$ which is produced here.
