Given a m x n rectangle, how many squares are there in it?
Input: m = 2, n = 2 Output: 5 There are 4 squares of size 1x1 + 1 square of size 2x2. Input: m = 4, n = 3 Output: 20 There are 12 squares of size 1x1 + 6 squares of size 2x2 + 2 squares of size 3x3.
Let us first solve this problem for m = n, i.e., for a square:
For m = n = 1, output: 1
For m = n = 2, output: 4 + 1 [4 of size 1×1 + 1 of size 2×2]
For m = n = 3, output: 9 + 4 + 1 [4 of size 1×1 + 4 of size 2×2 + 1 of size 3×3]
For m = n = 4, output 16 + 9 + 4 + 1 [16 of size 1×1 + 9 of size 2×2 + 4 of size 3×3 + 1 of size 4×4]
In general, it seems to be n^2 + (n-1)^2 + … 1 = n(n+1)(2n+1)/6
Let us solve this problem when m may not be equal to n:
Let us assume that m <= n
From above explanation, we know that number of squares in a m x m matrix is m(m+1)(2m+1)/6
What happens when we add a column, i.e., what is the number of squares in m x (m+1) matrix?
When we add a column, number of squares increased is m + (m-1) + … + 3 + 2 + 1
[m squares of size 1×1 + (m-1) squares of size 2×2 + … + 1 square of size m x m]
Which is equal to m(m+1)/2
So when we add (n-m) columns, total number of squares increased is (n-m)*m(m+1)/2.
So total number of squares is m(m+1)(2m+1)/6 + (n-m)*m(m+1)/2.
Using same logic we can prove when n <= m.
So, in general,
Total number of squares = m x (m+1) x (2m+1)/6 + (n-m) x m x (m+1)/2 when n is larger dimension
Using above logic for rectangle, we can also prove that number of squares in a square is n(n+1)(2n+1)/6
Below is implementation of above formula.
Count of Squares is 20
Alternate Solution :
Let us take m = 2, n = 3;
The number of squares of side 1 will be 6 as there will be two cases one as squares of 1-unit sides along the horizontal(2) and second case as squares of 1-unit sides along the vertical(3). that give us 2*3 = 6 squares.
When the side is 2 units, one case will be as squares of side of 2 units along only one place horizontally and second case as two places vertically. So number of squares=2
So we can deduce that
Number of squares of size 1*1 will be m*n
Number of squares of size 2*2 will be (n-1)(m-1)
So like this the number of squares of size n will be 1*(m-n+1)
The final formula for total number of squares will be n*(n+1)(3m-n+1)/6
Count of Squares is 20
Thanks to Pranav for providing this alternate solution.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
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