# Counting overlapping figures

How many four-sided figures appear in the diagram below?

I tired counting all the rectangles I could see, but that didn't work. How do I approach this?

• Just count it!:D – Mahdi Jul 20 '14 at 11:46
• In what sense did "counting all the rectangles" not work? Do you feel that you miscounted? (In which case: try again more carefully.) Or do you feel that you counted correctly, but seem to be counting the wrong things? (In which case: reconsider your definition of "side"; e.g. may sides overlap?) – Rebecca J. Stones Jul 20 '14 at 11:55
• I was looking for a systematic way to solve the problem. And I miscounted because the answer is 25 (I should have included that in the question). I am having a hard time seeing all the four-sided figures. – Guest Jul 20 '14 at 12:03
• Go through each corner, and count how many rectangles have that corner as a top-left vertex. That's the systematic way of doing it. – Arthur Jul 20 '14 at 12:12
• That is exactly what I was looking for Arthur. How can I choose your answer? – Guest Jul 20 '14 at 12:36

Go step by step.

First Picture: 1 rectangle

Second Picture: 2 additional rectangles. The small rectangle, which has been added and the big one, which contains the two small rectangles.

Third picture: The big rectangle. Then two rectangles, which contains 2 small linked rectangles. And the small rectangle, which has been added

Fourth picture: Only one small rectangle.

Fifth Picture: The rectangle, which contains the two small rectangle and the small additional rectangle.

You go on like this. Then sum the amount of rectangles.

Each rectangle has two vertical lines and two horizontal lines.

There are five vertical lines in the picture, we can label them 1, 2, 3, 4, 5.

If the leftmost edge is 1: Then the top and bottom are uniquely determined, and it is easy to see that 3 or 4 must be the right edge. 2 options.

If the leftmost edge is 2: Then the rightmost edge is 3 or 4 (2 choices), and in either case there are 3 horizontal segments that can serve as the top/bottom($\binom{3}{2} =3$ choices). So this gives $2 \cdot 3 = 6$. 6 options.

If the leftmost edge is 3: If the rightmost edge is 5 there is only one rectangle. If the rightmost edge is 4, there are 5 horizontal segments for top and bottom, so $\binom{5}{2} = 10$ choices. Hence 11 options.

If the leftmost edge is 4: Then the rightmost edge is 5, and there are four horizontal segments yielding $\binom{4}{2} = 6$ possible rectanges. 6 options

The total is 2 + 6 + 11 + 6 = 25.

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