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 Squares and Rectangles II (Posted on 2015-01-16)
(A) How many squares are there in the diagram given below?
(B) How many rectangles are there in the diagram given below?
```+-----+-----+-----+-----+
|     |     |     |     |
|     |     |     |     |
+-----+-----+-----+-----+
|     |     |     |     |
|     |     |     |     |
+-----+-----+-----+-----+
|     |     |     |     |
|     |     |     |     |
+-----+-----+-----+-----+
|     |     |     |     |
|     |     |     |     |
+-----+-----+-----+-----+
|     |     |     |
|     |     |     |
+-----+-----+-----+
|     |     |
|     |     |
+-----+-----+
|     |
|     |
+-----+```
*** If the length and width appear equal, then it is a square. If not, then it is a rectangle.

** The number of rectangles should exclude the ones that are counted as squares

 No Solution Yet Submitted by K Sengupta No Rating

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 A quick count. | Comment 1 of 2

The number of squares with the smallest square at its upper left corner:

```+-----+-----+-----+-----+
|  4  |  3  |  2  |  1  |
|     |     |     |     |
+-----+-----+-----+-----+
|  3  |  3  |  2  |  1  |
|     |     |     |     |
+-----+-----+-----+-----+
|  2  |  3  |  2  |  1  |
|     |     |     |     |
+-----+-----+-----+-----+
|  1  |  2  |  2  |  1  |
|     |     |     |     |
+-----+-----+-----+-----+
|  1  |  2  |  1  |
|     |     |     |
+-----+-----+-----+
|  1  |  1  |
|     |     |
+-----+-----+
|  1  |
|     |
+-----+```

For a total of 40

Number of rectangles with each small square at its upper left corner.  (Including squares. Subtract the above numbers to exclude squares.)

```+-----+-----+-----+-----+
|  16 |  15 |  12 |  7  |
|     |     |     |     |
+-----+-----+-----+-----+
|  12 |  12 |  10 |  6  |
|     |     |     |     |
+-----+-----+-----+-----+
|  8  |  9  |  8  |  5  |
|     |     |     |     |
+-----+-----+-----+-----+
|  4  |  6  |  6  |  4  |
|     |     |     |     |
+-----+-----+-----+-----+
|  3  |  4  |  3  |
|     |     |     |
+-----+-----+-----+
|  2  |  2  |
|     |     |
+-----+-----+
|  1  |
|     |
+-----+```

Interesting structure there.

Numbers sum to 155.

Subtract the 40 above to exclude the squares and get 115.

 Posted by Jer on 2015-01-16 09:03:48

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