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 nn equal pieces, with the same area and the same shape. Is it possible to divide such a triangle into 55 equal pieces? Is it possible to do it for an arbitrary value of nn?


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 n2n^2 pieces, and then either into 2n2,3n22n^2, 3n^2 and 6n26n^2. Any hints as to how I can either do it, or prove there is no way?

Note by Alexandre Miquilino
1 year, 10 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. 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 1

paragraph 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

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

Anirudh Sreekumar - 1 year, 10 months ago

Log in to reply

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!

Alexandre Miquilino - 1 year, 10 months ago

Log in to reply

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

Frost Constantin - 1 year, 9 months ago

Log in to reply

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

U K Swamy - 1 year, 8 months ago

Log in to reply

4 falls in the n2n^2 category, but thanks for the input.

Alexandre Miquilino - 1 year, 8 months ago

Log in to reply

oh sorry

U K Swamy - 1 year, 7 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...