$3$, $4$ and $5$ are on diagonals, so can't go in the centre square (since each number must appear just once on each diagonal).

So the number in the centre sqaure must be a $1$ or a $2$. Suppose it was a $1$. Then we must put $1$s as shown in red:

and $2$s as shown in blue:

So there must be a $1$ in place of one of the crosses:

But the top and bottom cross couldn't be $1$, since they are on the diagonal, and the middle cross can't be $1$ since it's on the $3$rd row (where there's already a $1$). So $1$ can't go in the centre square after all!

$2$ goes in the centre square.

Alternatively, consider the bottom-left corner. $2$ and $3$ are in the same corner, $4$ is in the same row and $5$ is in the same diagonal. Therefore, this must contain a $1$.

Then, the centre is in the same diagonal as each of the corners, so cannot be $1$, $3$, $4$ or $5$. This means the centre must contain a $2$.

So the number in the centre sqaure must be a $1$ or a $2$. Suppose it was a $1$. Then we must put $1$s as shown in red:

and $2$s as shown in blue:

So there must be a $1$ in place of one of the crosses:

But the top and bottom cross couldn't be $1$, since they are on the diagonal, and the middle cross can't be $1$ since it's on the $3$rd row (where there's already a $1$). So $1$ can't go in the centre square after all!

$2$ goes in the centre square.

Alternatively, consider the bottom-left corner. $2$ and $3$ are in the same corner, $4$ is in the same row and $5$ is in the same diagonal. Therefore, this must contain a $1$.

Then, the centre is in the same diagonal as each of the corners, so cannot be $1$, $3$, $4$ or $5$. This means the centre must contain a $2$.

*This problem is taken from the UKMT Mathematical Challenges.*