# 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
8 months, 3 weeks ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

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

- 8 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!

- 8 months, 3 weeks ago

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

- 6 months, 2 weeks ago

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

- 6 months, 2 weeks ago

oh sorry

- 5 months, 1 week ago

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

- 7 months, 2 weeks ago