Operation Z

The numbers 1 , 2 , 3 , , 20 1,2,3, \ldots , 20 are written on a blackboard.

We define an operation Z Z as: Erasing two numbers a a and b b and writing new number a + b 1 a+b-1 on a blackboard.

What number can be there on the blackboard after 19 19 operations?


The answer is 191.

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2 solutions

Akash Shah
May 29, 2014

For any collection of n n numbers on the board, let X X denote the sum of all of the numbers decreased by n 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 X = S + a + b - n , and after the transformation, we have X = S + ( a + b 1 ) ( n 1 ) = S + a + b n X = S + (a + b - 1) - (n - 1) = S + a + b - n . Thus, X is invariant. Initially, we have X = ( 1 + 2 + + 20 ) 20 = 19 20 2 = 190 X = (1 + 2 + · · · + 20) - 20 =\frac { 19\cdot 20 }{ 2 } = 190 . When there is only one number left, we must have X = 190 X = 190 , so the last number must be 191 191 .

Oliver Daniel
Apr 18, 2015

I got it using Excel

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