There are 36 ways.
There are many methods of counting. Here is one:
Consider the following diagrams, each of which is made up of 2x1 rectangles.
1. _ _ 2. _ _ 3. _ _ _ _
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The first diagram shows 1 of the 2 ways (the second way is the same, rotated 90 degrees) to make a 2x2 square. The second diagram shows the only way of making a 4x2 rectangle that cannot be broken up into smaller 2x2 squares. The third diagram shows 1 of the 2 ways (again, the second way is a rotation) to make a 4x4 square that cannot be broken up into smaller 4x2 or 2x2 rectangles.
Given these three basic units, we only need to break the 4x4 square into 2x2, 4x2, and 4x4 pieces. There are few enough ways that I can show all of them:
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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|_ _|_ _| |_ _ _ _| |_ _|_ _| |_ _ _ _|
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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|_ _ _ _| |_ _|_ _| |_ _| | | |_ _|
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|_ _|_ _| |_ _ _ _| |_ _|_ _| |_ _|_ _|
Though I only count 8 ways, I haven't taken into account the fact that the 4x4 and 2x2 squares can each be rotated. For example, the first square shown actually represents 16 ways, since each of the 4 2x2 parts has two possibilities.
Taking into account these extra possibilities, there are a total of __36__ (16+2+1+1+4+4+4+4) different ways. |