Counting Rectangles Formed from N Unit Squares: A Comprehensive Guide

Understanding How to Form Rectangles Using Unit Squares

To determine how many rectangles can be formed using N squares of unit length, we need to understand the fundamental properties of rectangles and how they can be constructed from unit squares. This guide will walk you through the step-by-step process, from understanding the basic configuration of unit squares to calculating the total number of rectangles that can be formed.

Step-by-Step Analysis

Understanding Rectangles

A rectangle can be formed by choosing two horizontal lines and two vertical lines. When considering N unit squares arranged in a grid, we can think of these squares as lying within a grid defined by horizontal and vertical lines.

Grid Formation

For N unit squares, if we assume they are arranged in a rectangular formation, the grid will have dimensions m x k, where m x k N. This means there will be (m-1) horizontal lines and (k-1) vertical lines.

Counting Lines

- The number of ways to choose 2 horizontal lines from (m-1) lines is given by the binomial coefficient C(m-1, 2).

- Similarly, the number of ways to choose 2 vertical lines from (k-1) lines is given by the binomial coefficient C(k-1, 2).

Total Rectangles

The total number of rectangles that can be formed is the product of the binomial coefficients for the horizontal and vertical lines:

Total Rectangles C(m-1, 2) x C(k-1, 2)

All Possible Configurations

Since N can be expressed as m x k for various pairs (m, k), we need to consider all factor pairs of N. For each factor pair, we compute the total number of rectangles.

Example Calculation

Let's consider N 6 as an example. The possible pairs (m, k) are (1, 6), (2, 3), (3, 2), and (6, 1).

- For the pair (1, 6):

C(0, 2) x C(5, 2) 1 x 10 10

- For the pair (2, 3):

C(1, 2) x C(2, 2) 1 x 1 1

- For the pair (3, 2):

C(2, 2) x C(1, 2) 1 x 1 1

- For the pair (6, 1):

C(5, 2) x C(0, 2) 10 x 1 10

The total number of rectangles is the sum of all these values:

10 1 1 10 22

Conclusion

In general, the number of rectangles that can be formed using N unit squares is given by summing the rectangles formed by all pairs of factors of N. The calculation involves finding all pairs (m, k) such that m x k N and using the combination formulas for counting rectangles.