Fun with MS Paint

I was wondering about MS paint and stumbled upon certain interesting problems. Here they are.

Problem 1: Can you draw an equilateral triangle in MS Paint?

Problem 2: Can you draw a regular hexagon in MS Paint?

Problem 3: Can you draw a regular polygon having odd number of sides in MS Paint?

Rules: You cannot use any aid to make measurements of any kind (length or angle). You can only use MS Paint (obviously). You are not supposed to use the co-ordinates to make measurements of any kind (length or angle). However, you can use the co-ordinates for precise placement of your mouse pointer.

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.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered 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"

Math	Appears 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}`	$\boxed{123}$

Comments

It's impossible; drawing on MS Paint requires the vertices to be lattice points, and it's a well-known result that a regular polygon that is not a square cannot have their vertices on lattice points (using that if $a,b,c$ are rationals and $\cos a\pi = b, \sin a\pi = c$ , then $2a$ is an integer).

Of course, assuming that MS Paint's canvas is infinitely divisible (so it's basically a plane of reals times reals), then it's the usual compass and straightedge problem.

Ivan Koswara - 7 years, 7 months ago

Read my comment below. The equilateral triangle and regular hexagon can be constructed (and should looks okay if the screen resolution and the size of the computer screen is 1:1). Only the pentagon is impossible to construct.

Adam Zaim - 7 years, 6 months ago

In Windows 7, you can make a hexagon and a pentagon. Plus, you didn't say a REGULAR hexagon. So, problem 2 and 3 have been solved. But I don't know how to make equilateral triangle.