Problem number 23. Aria decides to paint each square in the shape shown so that squares painted the same color do not touch. She paints four squares red, three squares blue, two squares green, and one square yellow. Which color did she paint the square marked with the X? To solve this problem, let's divide the shape into several smaller shapes. For example, in this picture you can see that the large triangle has been divided into the square with the X and three smaller triangles in the corner, each of which contain three squares. For example, let's look at the triangle circled with a red circle. In this triangle of three squares, at most one of these squares can be red. This is because that if there are two red squares in this triangle, they have to touch, because any two squares in this triangle will touch. We can apply the same logic to all the other triangles in the picture. In the three triangles, there can be at most three red squares. But we know that there are four red squares in total, and in the three triangles, you can have at most three of them. That means that the fourth red square has to be in the position with the X, since that's the only square left that can possibly have a red square. This means the square marked X, the center square, must be red. So the answer is A, red. To prove to yourself that this works, you can try and figure out a possible way to fill out the squares using guess and check. The picture shows one way. As you can see, the center square is red. So the answer is A, red.

painting squares

coloring problem

adjacent squares

red square

triangle division