You are given N unit squares (squares with side length 1 unit), and you are asked to make rectangles using these squares. You have to count the number of rotationally unique rectangles than you can make. What does rotationally unique mean? Well, two rectangles are rotationally unique if one can’t be rotated to become equivalent to the other one.
Example – The 4×2 rectangle can be rotated 90 degrees clockwise to make it the exact same as the 2×4 rectangle and so these are not rotationally unique.
Input : N = 4 Output : 5 We can make following five rectangles 1 x 1, 1 x 2, 2 x 2, 1 x 3 and 1 x 4 Input : N = 5 Output : 6 Input : 6 Output : 8
So how do we solve this problem?
Every rectangle is uniquely determined by its length and its height.
A rectangle of length = l and height = h then l * h <= n is considered equivalent to a rectangle with length = h and height = l provided l is not equal to h. If we can have some sort of "ordering" in these pairs then we can avoid counting (l, h) and (h, l) as different rectangles. One way to define such an ordering is:
Assume that length <= height and count for all such pairs such that length*height <= n.
We have, length <= height
or, length*length <= length*height
or, length*length <= n
or, length <= sqrt(n)
This article is contributed by Hemang Sarkar. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
- Determine the number of squares of unit area that a given line will pass through.
- Smallest square formed with given rectangles
- Check if N rectangles of equal area can be formed from (4 * N) integers
- Number of rectangles in N*M grid
- Number of rectangles in a circle of radius R
- Find the minimum number of rectangles left after inserting one into another
- Count the number of rectangles such that ratio of sides lies in the range [a,b]
- Minimum number of squares whose sum equals to given number n
- Find maximum number that can be formed using digits of a given number
- Find the number of rectangles of size 2*1 which can be placed inside a rectangle of size n*m
- Paper Cut into Minimum Number of Squares
- Number of ways of writing N as a sum of 4 squares
- Count number of squares in a rectangle
- Number of perfect squares between two given numbers
- Check whether a number can be represented by sum of two squares
Improved By : jit_t