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For example, I stated this fact/theorem at a dinner to some friends and they were pretty impressed.

Given any sequence of n integers, positive or negative, not necessarily all different, some consecutive subsequence has the property that the sum of the members of the subsequence is a multiple of n.

wording taken from http://www.cut-the-knot.org/pigeonhole/group.shtml

I stated the above in laymans terms so they could understand it and they were interested in it (at least for a few minutes).

What are some other accessible theorems/facts similar to this that non-mathematicians can understand? or maybe simple applications of mathematics that they will understand?

Cheers.

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Far too many to list! Here are some things that sprang to mind.

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    $\begingroup$ The cylindrical glass question might also be phrased like this: I leave my house at 8 AM and walk to my grandmother's house, arriving at 8 PM. The next day I say goodbye at 8 AM and depart, arriving home by the same route at 8 PM. There is at least one point during the journey when I was at the exact same place at the exact same time of day on the two days. $\endgroup$ – MJD Aug 23 '13 at 16:21
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Stir a cup of tea, but don't let your spoon touch the walls of the cup. Once the tea has settled, there is some point in the liquid which is in exactly the same place in the cup as before you stirred it.

This is a consequence of the Brouwer Fixed Point Theorem.

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    $\begingroup$ Is stirring tea really going to be a continuous function? I usually describe this theorem as cutting a circle from a piece of paper, wrinkling & stretching it, and then pressing it back into the hole (it tends to be more fun as most people will then try to "break" the theorem by describing ways of doing it, and you can help them find the fixed point in each one). $\endgroup$ – MartianInvader Aug 23 '13 at 16:32
  • $\begingroup$ Well, if you have a suitably viscous tea... $\endgroup$ – user1729 Aug 26 '13 at 8:04
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The Ham sandwich theorem is fun. If you have something (a ham sandwich) consisting of three pieces of any shape, then it's always possible to cut it with one single straight cut into two ham sandwiches, in such a way that all three pieces are equally divided between the sandwiches.

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Every sphere which sprouts hair has a quiff.

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I very much like Sylvester–Gallai theorem:

Given a finite number of points in the Euclidean plane, either

  1. All the points are collinear; or

  2. there is a line which contains exactly two of the points.

Kelly's proof of this theorem is a real mathematical gem.

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I think the proof that irrational numbers exist is easy to understand the fact is quite interesting.

Theorem: Not all numbers can be expressed as decimal fractions

Proof: Suppose you could write $\sqrt{2}$ as a completely shortened number, that means it exists $p,q \in \mathbb{N}$ such that

$\begin{align}\sqrt{2} &= \frac{p}{q}\\ \Leftrightarrow 2 &= \frac{p^2}{q^2}\\ \Leftrightarrow 2 q^2 &= p^2 \end{align}$

But this means that $p^2$ is even which means that $p$ is even. So we can write $p$ as $p = 2r$:

$\begin{align}2 q^2 &= p^2\\ \Leftrightarrow 2q^2 &= (2r)^2\\ \Leftrightarrow 2q^2 &= 4r^2\\ \Leftrightarrow q^2 &= 2r^2 \end{align}$

This means that $q$ is also even. So both, $p$ and $q$ have $2$ as a factor. This means $\sqrt{2}$ cannot be written as a completely shortened fraction. As every rational number can be written as a completely shortened fraction, $\sqrt{2}$ cannot be a rational number $\blacksquare$


Also very interesting:

  • Birthday paradox (which can easily be explained with the pigeonhole principle)
  • Seven Bridges of Königsberg: Eulers Theorem (A graph has a Eulerian cycle $\Leftrightarrow$ Every vertex has an even degree). You might also find interesting theorems in this presentation which I made as an introduction to graph theory.
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God's number is 20: Every position of Rubik's cube can be solved in at most 20 moves.

R(3,3)=6: In any party of six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances.

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Take a beachball and deflate it, then flatten it against the ground, as wrinkled and stretched out as you like (but not torn). There will then always be two opposite points of the beachball pressed against the same point on the ground (the Borsuk-Ulam theorem).

Another fun way to phrase the theorem is that at any given time, two completely opposite ends of the earth have the exact same temperature and pressure.

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  • $\begingroup$ For temperature and pressure, why should the pair of antipodal points be the same? $\endgroup$ – Alex R. Aug 23 '13 at 16:48
  • $\begingroup$ @AlexR. Temperature and pressure are both 1-dimensional values. The map from the surface of the earth to the pair (temperature, pressure) is a map $S^2 \rightarrow \mathbb{R}^2$, thus has a pair of antipodal points with the same values in $\mathbb{R}^2$, thus both the same temperature and same pressure. $\endgroup$ – MartianInvader Aug 23 '13 at 17:51

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