**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:

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

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