(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

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