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# In dividing a equilateral triangle into n equal parts with equal areas and equal shapes...

People of Brilliant, how are you all? I came across an interesting puzzle that I want some help solving.

Suppose we want to divide an equilateral triangle into $$n$$ equal pieces, with the same area and the same shape. Is it possible to divide such a triangle into $$5$$ equal pieces? Is it possible to do it for an arbitrary value of $$n$$?

My investigations showed me that 2 and 3 are the most basic divisions available, and then onward I could find ways to divide a triangle into $$n^2$$ pieces, and then either into $$2n^2, 3n^2$$ and $$6n^2$$. Any hints as to how I can either do it, or prove there is no way?

Note by Alexandre Miquilino
2 months, 4 weeks ago

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If the pieces needn't be contiguous, it turns out it can be done as shown here !!

- 2 months, 3 weeks ago

While this is not exactly a solution I wanted, there is a reference to a paper in another answer which seems to state it is impossible. Thanks for the find!

- 2 months, 3 weeks ago

4 join all the midpoints of the sides forming 4 equilateral triangles each of equal area

- 2 weeks, 5 days ago

4 falls in the $$n^2$$ category, but thanks for the input.

- 2 weeks, 5 days ago

Hi,may I ask that where do you live,which country?

- 1 month, 2 weeks ago