How many rectnagles are there?
How many rectangles can we draw in an 8 × 8 chocolate along the grid and/or perimeter?
Clarification:
- The rectangles can be of any size and shape, which includes equilateral rectangles.
-
The rectangles can overlap.
The answer is 1296.
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2 solutions
Choose a coordinate for one vertex of the rectangle. There are 9 × 9 vertices to choose from.
Choose a coordinate for the opposite vertex of the rectangle. For the rectangle to not be a flat line, there are now ( 9 − 1 ) × ( 9 − 1 ) vertices to choose from.
Since each rectangle has 4 vertices, we need to divide by 4 to eliminate repeated rectangles.
This gives a total of 4 9 ⋅ 9 ⋅ ( 9 − 1 ) ⋅ ( 9 − 1 ) = 1 2 9 6 possible rectangles.
Also, it is possible to solve this problem by considering the number of horizontal and verticals lines. There are 9 horizontal lines and 9 vertical lines and any rectangles can be formed from the intersection of any TWO horizontal lines with any TWO verticals lines. In the case, there are 9 C 2 horizontal combinations and 9 C 2 verticals combinations. Then the total number of Rectangles = 9 C 2 × 9 C 2 = 1 2 9 6 rectangles.
( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ) 2 = 1 2 9 6