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I did not see that a student had posted a response before I started typing my answer (after having taken the time to solve the problem)! I always enjoy these mental spatial arrangement games.
The key for the first problem is to realize that the new triangles are necessarily going to be smaller than the original diamonds. This means that the new triangles are going to most efficiently made by cutting existing diamonds in half. Since we can only move four pencils, remove the left and rightmost two pencils. Use two of them to cut the two remaining diamonds in half with horizontal lines to make four triangles. Then use the other two to make two vertical lines on the sides to make two new triangles. I do not have the ability of upload an image right now but I have used the keyboard to make a rough drawing below:
The key to the second problem is to keep in mind that the two different squares do not need to be the same size. This means that the top four inside pencils can be removed to create one large square at the top with two pencils per side. All that remains is to remove two of the outside pencils on the bottom to leave one small square remaining with one pencil per side. Again, a rough drawing is below:
In the image, the green sticks are moved/removed.
In the first problem, a horizontal orientation is also valid (i.e. moving of the top two and bottom two matchsticks).
In the second problem, any combination that results in 2x2 square and one 1x1 square (not inside the 2x2) is valid.
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