Dividing a Square into N smaller squares | Puzzles for Interview - Interview Preparation PDF Download

Introduction

This article aims to explain how to find all values of N for which a square can be divided into N smaller squares. We will also outline an algorithm for performing such a dissection.

Important Observation

A square has 4 right angles. To divide a square into smaller squares, each of its right angles must fall into another square. Otherwise, non-square figures will be formed.

Case 1: N = 2, 3, or 5

In this case, dividing a square into 2, 3, or 5 smaller squares is not possible, as it doesn't satisfy the right-angle condition mentioned above.

Case 2: N = 4

This is the simplest case. Simply divide the square horizontally and vertically from the center. The resulting figure will have 4 smaller squares.

Case 3: N is even and greater than 4

For this case, we can generalize the solution by considering N = 2k. We can form 2k - 1 equal squares along adjacent sides of the given square. The side length of each smaller square should be equal to 1/k of the length of the given square. For example, when N = 6, we form 5 squares along the top and right side, each with a side length equal to 1/3 of the original square's side length. This leaves a square with a side length of 2/k, resulting in a total of 6 squares.

Case 4: N is odd and greater than 5

This case is based on the solution for even values of N. If N is odd, we can express it as N = 2k + 1, which can also be written as N = 2(k - 1) + 3. First, we form 2(k - 1) squares using the approach described in Case 3. Then, we divide one of the obtained squares into four smaller squares, increasing the overall square count by 3. For example, when N = 9, we first form 6 squares, then divide the top-left square into 4 smaller squares, resulting in a total of 9 squares.

Conclusion

By considering these cases, we can find all possible values of N for which a square can be divided into N smaller squares and outline an algorithm for performing such a dissection.