Question

How many rectangles does a $4\times 4$ square has?

• A. $1^3+2^3+3^3+4^3$
• B. $1^3+2^3+3^3$
• C. $1^3+2^3+3^3+4^3+5^3$
• D. $1^3+2^3+3^3+4^3+5^3+6^3$

Answer 1

The correct option is A $1^3+2^3+3^3+4^3$


Along the top row of the $4\times 4$ square,
Total number of rectangle of size $1\times 1=4$
Total number of rectangle of size $1\times 2=3$
Total number of rectangle of size $1\times 3=2$
Total number of rectangle of size $1\times 4=1$
Total Number of rectangles in first row = 10
Now, along the first column,
Total number of rectangle of size $1\times 1=4$
Total number of rectangle of size $1\times 2=3$
Total number of rectangle of size $1\times 3=2$
Total number of rectangle of size $1\times 4=1$
Total Number of rectangles in first column = 10
Hence the total number of rectangle in $4\times 4$ square grid $=10\times 10={10}^2$
Now ${10}^2=1^3+2^3+3^3+4^3$
Therefore, total no of rectangles in $4\times 4$ square grid $=1^3+2^3+3^3+4^3$

Alternative Solution:

We know that $n\times n$ square has $1^3+2^3+3^3+.....+n^3$ rectangles
Similarly, $4\times 4$ square grid has $1^3+2^3+3^3+4^3$ rectangles.