It's a bit hard to explain without the drawings, but I'll try. We know that it's a square with an area of 100, so each side needs to be 10. They ask how many such squares can be put on the coordinate plane so that all the points are integers, when one of them is on the origin, or in other words, how many triangles we can put on the plane, when one vertices is on the origin and the hypothesis is 10? And how many squares are just parallel to the axis?
First, look on the 2nd type. We have 4 different squares: one on each quarter when the 2 sides of each lie on the x and y axis.
Now, let's see how can we rotate the square in the 1st quarter to the side so that the coordinates of the point created are integers? What right triangles will give us a hypothesis of 10 while the sides being integers? x^2+y^2=10^2
We can see that this is a 3:4:5 right triangle, so x can be either 6 or 8, and so is y. Therefore, we have two triangles that meet the restriction, thus 2 squares.
However, this is just in one quarter, but we have 4 of these, so we have 8 squares. Add to this the 4 we have found before, and we have 12.
Is it E?
Leon
