Operation Z
The numbers are written on a blackboard.
We define an operation as: Erasing two numbers and and writing new number on a blackboard.
What number can be there on the blackboard after operations?
The answer is 191.
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For any collection of n numbers on the board, let X denote the sum of all of the numbers decreased by n . If the sum of all the numbers except a and b is equal to S, then before the transformation, we have X = S + a + b − n , and after the transformation, we have X = S + ( a + b − 1 ) − ( n − 1 ) = S + a + b − n . Thus, X is invariant. Initially, we have X = ( 1 + 2 + ⋅ ⋅ ⋅ + 2 0 ) − 2 0 = 2 1 9 ⋅ 2 0 = 1 9 0 . When there is only one number left, we must have X = 1 9 0 , so the last number must be 1 9 1 .