Dominoes

Note that as each domino takes up 2 squares then $n$ must be even

Now take an even $n\times n$ board and color it as in figure 1. Note that every vertical domino will take up 1 and only 1 shaded square while every horizontal domino will take up either 0 or 2 shaded squares. As there are an even number of shaded squares then we must have an even number of vertical dominoes. Say we have $2k$ vertical dominoes.

Now as the number of vertical dominoes equals that of horizontal dominoes we must have $2k$ horizontal dominoes. Thus we have a total of $4k$ dominoes and as each domino takes up 2 squares we have a total of $8k$ squares taken up. Therefore the number of squares in the board must be a multiple of 8.

Thus $n^2$ must be a multiple of 8 so $n$ must be a multiple of 4. Now if $n$ is a multiple of 4, say $n=4m$ for some positive integer $m$ then we can divide the board into $m^2$ $4\times 4$ boards each which can be tiled with 4 horizontal and 4 vertical dominoes as shown in figure 2. So we can achieve the condition if and only if $n$ is a multiple of 4.

So our answer is all the multiples of 4 from 1 to 100 $\implies \fbox{25}$