# Puzzle | Dividing a Square into N smaller squares

Puzzle: Find all values of N for which one can dissect a square into N smaller squares, and outline an algorithm for doing such a dissection.

Solution: The basic point to observe is a square has 4 right-angles. So, to divide it into smaller squares each of its right-angle must fall into another square, as more than one right-angle together will result in a non-square figures.

Now, consider the following cases:

1. When N = 2, 3, or 5: No such division is possible, as it violates the above given condition and non-shaped figures are obtained.

2. When N = 4: This is the easiest case. Just divide the square horizontally and vertically, from the centre. The resulting figure will have 4 squares.

3. When N is even and greater than 4: This case can be generalised by considering N = 2k and forming 2k – 1, equal squares along adjacent sides of the given square. However, the side length of each smaller square should be equal to 1/k of the length of the given square.

For example: Consider the example when N = 6 as shown in the figure, here we have formed 5 squares along the top and right-side, each of side (1/3)rd of the side of the original square. Also, a square of side (2/k) is left, resulting in a total of 6 squares.

4. Case N is odd and greater than 5: This case builds upon the solution for even values of N. If N is odd, we can break it as N = 2k + 1, which further can be written as N = 2(k – 1) + 3. Now, we can first form 2(k – 1) squares using the above approach, and then divide, on of the obtained squares, into four smaller squares, which will increase the overall square count by 3.

For example: Consider the example when N = 9 as shown. Here, we first form 6 squares, and then divided the top-left square into 4 smaller squares, to get total 9 squares.

GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :
Practice Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.