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I need a formula to divide a fixed area in 'N' number of equal parts. For example I am having a picture/image of 800*800, So now I need to slice it into 10 number of equal parts (squares/ rectangles) and what will be the dimension (height * width) of the sliced portions. Can you help me in that? If in case it is an odd number then we will add +1 to it and will merge the last 2 images resulting in making a rectangle.

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Dimension of the sliced portions will depend on how you slice it. You might slice out 3 ciecles, 5 squares, and other 5 not definable with fewer than 100 words. But however area of each slice is quite clear (6400 sq.units). – user21436 Feb 7 '12 at 8:01
Hello , each time I need to divide it into Squares/ rectangles – user1129951 Feb 7 '12 at 8:06
What happens if $N = 3$? – Sp3000 Feb 7 '12 at 8:11
Also, I'm a bit confused as to why the trigonometry tag is there... – Sp3000 Feb 7 '12 at 8:20

Let's assume that $N$ divides $800 \times 800 = 64000$ for now. Then in order to split up the $800 \times 800$ image we factorise $N$ into integers $(a, b)$ such that $a|800$ and $b|800$.

For example, if $N = 10$ then you have $(1, 10)$, $(2, 5)$, $(5, 2)$ and $(10, 1)$ as valid factorisations. Then each rectangular slice would have dimensions $\frac{800}{a} \times \frac{800}{b}$. So for $N = 10$ the solutions are:

  • $\frac{800}{1} \times \frac{800}{10} = 800 \times 80$

  • $\frac{800}{2} \times \frac{800}{5} = 400 \times 160$

  • $\frac{800}{5} \times \frac{800}{2} = 160 \times 400$

  • $\frac{800}{10} \times \frac{800}{1} = 80 \times 800$

Note that for some $N$, there may be. For example, $(1600, 1)$ is not a valid factorisation for $N = 1600$ because 1600 exceeds the side lengths of the initial $800 \times 800$ image.

If $N$ does not divide 64000 then you cannot divide the image into equal rectangles.

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