Count 'em All 8!

Probability Level 3

Count the number of 3 × 6 3\times6 rectangles in the 50 × 39 50\times39 grid above.

I'm lazy to draw new grids by the way.

Details and assumptions:

  • Dimensions are expressed as width × \times height . Thus a 3 × 6 3\times6 rectangle (that is, a rectangle of width 3 and height 6) is considered to be different to a 6 × 3 6\times3 (width 6 height 3) rectangle, in this case, you only want the 3 × 6 3\times6 rectangles, not the 6 × 3 6\times3 ones.
This is one part of Quadrilatorics .


The answer is 1632.

Kenneth Tan by Kenneth Tan , 21, Malaysia

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2 solutions

Kenneth Tan
Mar 17, 2016

From this note , we know that the number of x × y x\times y rectangles in an a × b a\times b grid given that x ⩽ a x\leqslant a and y ⩽ b y\leqslant b is ( a − x + 1 ) ( b − y + 1 ) (a-x+1)(b-y+1)

Now, substitute a = 50 a=50 , b = 39 b=39 , x = 3 x=3 , y = 6 y=6 into the equation above we would get 1632 1632 .

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Ashish Menon
Mar 25, 2016

In a figure, the number of ( a × b ) (a×b) rectangles in a ( x × y ) (x×y) figure , where x ≥ a x \geq a and y ≥ b y \geq b can be found by the formula :- ( x − ( a − 1 ) ) ( y − ( b − 1 ) ) (x -(a-1))(y-(b-1))
= ( 39 − ( 6 − 1 ) ) ( 50 − ( 3 − 1 ) ) (39 -(6-1))(50-(3-1))
= ( 39 − 5 ) ( 50 − 2 ) (39 -5)(50-2)
= ( 34 × 48 ) (34×48)
= 1632 1632 rectangles. □ _\square



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