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Take a look at these two images.

Can you draw them by connecting the dots:

a) WITHOUT lifting your pencil,

b) WITHOUT retracing any stroke, and

c) Starting and ending at the same dot?

This thread will help you solve puzzles like this.

(h/t @10kdiver for collaboration!) 2/

The image on the right CAN be drawn without lifting our pencil, following all the rules above.

Here's one way to do it:

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Math is amazing.

Consider the following famous question:

"Is the square root of 2 rational: can it be expressed as a ratio of integers?"

This thread answers our question using an unconventional approach: the Euclidean Algorithm, which is at the core of number theory. 2/

We all know that the answer to our question is "no." Sqrt(2) is irrational.

The typical proof of this statement uses an algebraic method (see below for outline of the proof).

But can we find a more elegant, geometric method to prove the irrationality of sqrt(2)?

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Math is amazing.

Here’s a trick question that will take us deep into probability theory:

Suppose that I give you a circle, and ask you to pick a point A in the circle at random.

What’s the probability that A is closer to the outside of the circle than it is to the center? 2/

Well, first we’ll need to define some lengths and points.

Let O be the point that is the CENTER of the circle.

Let C be our circle.

And let R be the radius of the circle (R > 0).

(As a reminder, A is the random point that we are selecting in the circle)

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Math is amazing.

Pascal's Triangle is one of the core ideas of math.

It's used in probability, combinatorics and algebra.

In this thread, you'll learn about how this triangle can be applied to counting paths, the choose coefficients, and the binomial theorem. 2/

The setup of this problem starts with simplicity, as is often common in math.

We will start with an infinitely large city named Euclidtown.

This city has perfectly straight streets, all intersecting at 90˚ angles.

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Math is amazing.

In this thread, I'll guide you through the double angle identities involving sine and cosine.

I will explain these identities using a "proof without words," similar to this figure.

See below for figure. 2/

Before we dive into this complicated figure, let's review what sin, cos, and tan even mean!

There are many ways to interpret these expressions. The simplest, most useful idea is that when we scale a triangle without changing its angles, ratios of the sides don't change!