What's your favorite proof accessible to a general audience? [closed]

What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in roughly $5 \pm\epsilon$ minutes. Let's define 'general audience' as approximately an average adult with education and experience comparable to someone holding a bachelor's degree in any non science major (e.g. history) from an average North American university.

I really like the proof of $$\sum_{i=1}^n i = \dfrac{n(n+1)}{2}$$ in which $1 + 2 + \cdots + (n-1) + n$ is written forwards then backwards and summed. It is claimed that Gauss had come up with this when he was just a child, although contested.

The proof

Let $$s = 1 + 2 + \cdots + (n-1) + n.$$ Clearly, $$ s = n + (n-1) + \cdots + 2 + 1.$$

Sum to get $$2s = \underbrace{(n+1) + (n+1) + \cdots + (n+1) + (n+1)}_{n \text{ times}}.$$

Hence, $$2s = n(n+1),$$ and $$s = \dfrac{n(n+1)}{2}.$$

I'm a bit reluctant to throw another answer on the pile, especially because I think there are other lists on this website which serve a pretty similar purpose. However, I think an excellent 5 minute blurb could be given to a general audience on the trick, attributed to von Neumann, for performing a fair coin toss when only a biased coin with unknown bias is available. There is a wikipedia entry on this. Here is an informal description:

Suppose you have a biased coin. The chance that the coin comes up heads or tails (assume both are actually possible) are unknown to you, but do not change from toss to toss. You and your friend Jane wish to use this coin to decide (fairly) which of you gets the top bunk at math camp.

Jane makes the following observation. Suppose the coin is flipped twice in a row. The possible outcomes are: \begin{align*} HH && TT && HT && TH \end{align*} Now, we do not know how likely each of these outcomes is, but one thing is certain: $$ \text{ The outcomes $HT$ and $TH$ are equally likely.}$$ Because of this observation, you agree on the following fair way to settle the dispute. You "call" the outcome $HT$ while Jane "calls" the outcome $TH$, and then proceed to flip the coin twice. If either $HH$ or $TT$ occurs, a mistrial is declared and you start over, flipping the coin another two times. Eventually, either $HT$ or $TH$ will occur, and the dispute is settled.

I think this works well because:

• It is simple -- simple enough to fit into 5 minutes.
• It requires no special knowledge from the audience. People generally have pretty reasonable built-in intuition for probability.
• It is a beautiful, but also practical, idea -- making it effective as "Math P.R."
• You can also actually demonstrate the procedure, to make sure it is understood, without any special equipment. Just take something such as a thimble which can land either of two states, but for which it is not clear that either outcome is equally likely.

The harmonic series diverges because otherwise there exists a finite number \begin{align*} S &= 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\dotsb \\ &= \left(1+\frac{1}{2}\right)+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}\right)+\dotsb \\ &> \left(\frac{1}{2}+\frac{1}{2}\right)+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{6}+\frac{1}{6}\right)+\dotsb \\ &= 1+\frac{1}{2}+\frac{1}{3}+\dotsb \\ &= S \end{align*}

I'd probably go for Euclid's beautiful proof of the infinitude of primes, as it doesn't require much knowledge beyond elementary school.

Edit: Another possibility might be the pictorial proof that the derivative of $x^2$ is $2x$ by a square whose side length, $x$, increases and what that means to the rate of change of the area

Existence of Eulerian walks and the whole 'seven bridges of Königsberg' story. It's a cliché, but it's not about numbers (which is a plus when talking to a general audience), and it's something people can find at least mildly amusing. The argument is simple and everyone can follow it, and the best thing is, it's not just a clever solution to what looks like a random puzzle, which I feel is the impression people can get when shown other simple proofs. Starting from the seven bridges perspective and replacing shores and islands with vertices and bridges with lines, this proof is the best toy example I know of of the power of mathematical abstraction.