$\underline{\text { Folklore }}$

Can some of the minuses in the expression $2013^{2}-2012^{2}-\ldots-2^{2}-1^{2}$ be replaced with pluses so that the value of the resulting expression equals 2013?
$(m+1)^{2}-m^{2}=2 m+1$. Therefore, $\left((n+3)^{2}-(n+2)^{2}\right)-\left((n+1)^{2}-n^{2}\right)=(2(n+2)+1)-(2 n+1)=2 n+5-2 n-1=4$. Thus, before the squares of any four consecutive natural numbers, signs can be placed in such a way that the value of the resulting expression is equal to 4.

We will divide the first 2012 squares into 503 such quadruples, and in each of them, we will place the signs in the specified manner. Before $1^{2}$, we will place

the "+" sign. The value of the resulting expression will be: $4 \cdot 503+1=2013$.

## Answer

It is possible.