Number 25. Julio wants to construct the cross shape shown in the picture using pieces shaped like the ones below the cross. He has many copies of each piece. The pieces can be rotated if needed, but they cannot overlap. What is the smallest number of pieces Julio could use to construct the shape? Pause if you want to try and solve it. Okay, so the key here is to figure out how to make these corners of the cross, the ends of the cross. Because for the middle, we can just use one square here, and then one of these 2x2 rectangles for each of these parts. And that is the best way that we can cover that. So you might think that we can get these edge parts by putting one of these corner pieces, or these triangle pieces, over here. Then a square here, and then another triangle. And same thing, a triangle, a square, and then another triangle. But we can actually do better than that. So if we erase this, we can actually use this piece over here, and this piece. It's a little hard to see, but if we put this piece over here, so that it looks like this. This sort of diamond-shaped piece. Then we can put this triangle over here. And now, instead of using 3 pieces to get this end part, we only use 2. And so we can do the same thing for each of the other ones. Put a diamond over there and a triangle, and then another triangle and a diamond, and then one last one. Now if we count the number of pieces, we have 1, 2, 3 on each of these sides of the cross, and then 1 in the middle. So that's 3 times 4 for the number of these side parts, plus 1 in the middle. And that's going to be 13. So the smallest number of pieces we can use is answer option C, 13.

cross shape

efficient assembly

diamond piece

13 pieces

optimization