Invariant

An invariant is a quantity or a property of an object that is unchanged under-considered operations.

Problem (Kyiv City, 1974)

Numbers $1$, $2$, … , $1974$ are written on a board. You are allowed to replace any two of these numbers with one number, which is either the sum or the difference of these numbers. Is it possible that after $1973$

times performing this operation, the only number left on the board is zero?

Solution

Answer: no, it is impossible.

Let us consider a quantity $Q$ being the sum of all written numbers on the board. In the beginning, the sum is

$$ Q = 1 + 2 + … + 1974 = \frac{1974 \cdot 1975}{2} = 987 \cdot 1975 $$

 

and thus $Q$ is an odd number. If the pair of numbers $(a,b)$ is changed to $a+b$, then the sum $Q$ does not change. If the pair of numbers $(a,b)$ is changed to $a-b$, then the sum $Q$ changes by an even number. Indeed

$$ (a+b) – (a-b) = 2b $$

Therefore the sum of all numbers on the board $Q$ is invariant modulo $2$ under-considered operations. Since in the beginning, $Q$ was odd and in the end, $Q$ is even, then we conclude that it is impossible for the last number to be zero.

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