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Squares and Rectangles II (Posted on 2015-01-16) Difficulty: 2 of 5
(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    
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Solution 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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