Number 25. What is the smallest number of colored squares that can be added so that this 6 by 6 square has 4 axes of symmetry? The 4 axes of symmetry are horizontal, vertical, diagonal, right, and left. Okay, so based on that, what can we do in order for all of these axes of symmetry to met, the 3-colored pattern must be in all sections created by this division? Well, let's just break it down first. If we only wanted to do this vertical axis of symmetry, then we would need this colored pattern to be on one side of the vertical symmetry and then also on the other side. Then we would need to add 3. Well, what about if it were just the horizontal? Then we would also just need to add 3 over here in this axis of symmetry. However, when we want both vertical and horizontal, now we have to add to take care of both the vertical, but also the horizontal and the mirror image of that. So we need to add green over here. We would need to add green over here. We would need to add green over here. Now, when we add these two more diagonals, that basically tells us that we need this 3-squared colored pattern to be there in every single one of these little triangles that's formed by these axes of symmetry. So in each of these 8 sections, each one should contain 3-colored squares. And we already have one section that's colored, so we do 8 times 3 minus 3 equals 21. The answer is 8. And this is super similar if, for example, you fold a piece of paper, something like this, and then you cut a piece out of it, you'll see that there's a hole in each of these 8 things. And that's because there are these 4 axes of symmetry. Same thing if you use a blotch of paint. You'll see the same thing happen over here. And that's kind of the intuitive explanation for why you get that these are 8 sections and each of them have to contain the same thing.

4 axes of symmetry

6x6 square

3-colored pattern

symmetrical uniformity

21 additional squares