Monovariant is a quantity related to the object that is monotonous under-considered operations.

Problem (Tournament of Towns, 2005)

Several stones are placed on an infinite (in both directions) strip of squares. As long as there are at least two stones on a single square, you may pick up two such stones, then move one to the preceding square and one to the following square. Is it possible to return to the starting configuration after a finite sequence of such moves?

Solution

Answer: no, it is impossible.

Let us number the squares as consecutive integer numbers. Let $x_1$, $x_2$, … , $x_n$ be the positions of the stones. Consider a quantity $Q$ to be the sum of squares of the positions of all stones on the strip:

$$ Q = x_1^2 + x_2^2 + … +  x_n^2 $$

 
Let two stones on the position $n$ were taken and placed on the positions $n-1$ and $n+1$ respectively. Before the operation, these two stones contributed to the quantity $Q$ the value of

$$ n^2 + n^2 = 2n^2 $$

After the operation these two stones contribute to the quantity $Q$ the value of

$$ (n-1)^2 + (n+1)^2 = 2n^2 + 2$$

After each operation the sum of squares of all positions $Q$ increases by $2$ and is monovariant. Therefore, it is impossible to return to the starting configuration after a finite sequence of such moves.

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