Skip to content
Related Articles

Related Articles

Improve Article
Puzzle | Dividing a Square into N smaller squares
  • Difficulty Level : Hard
  • Last Updated : 23 Apr, 2020

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.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

In case you wish to attend live classes with industry experts, please refer Geeks Classes Live

My Personal Notes arrow_drop_up
Recommended Articles
Page :