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As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like:

"You have a great circular pizza with $n$ toppings. Show that you can divide the pizza equitably among $k$ persons, which means every person gets a piece of pizza with exactly $\frac{1}{k}$ of any of the $n$ topping on it."

Have you more examples which are particular interesting or instructive?

EDIT: Since this is turning into a list of mathematical jokes or sophisticated proofs for simple facts, I may have to be more precise what I was asking for: a "real life example didactically used to motivate a mathematical theorem" (thanks to Lord_Gestalter for this great wording).

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Any application of the Hairy Ball Theorem is funny. –  David H May 15 at 18:55
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It is an application of comb-bin-a-torics. –  ncmathsadist May 15 at 18:55
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The Bible describes an early application of the Banach-Tarski paradox. –  Rahul May 15 at 18:59
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Too silly for an answer, but my Discrete Mathematics professor always starts the course with a lecture about Induction. As an example, he says that if students fit in the room on the n-th lecture, they will fit in on the (n+1)-th. The lecture was in a pretty tight room, and almost all seats were taken on the first lecture, but he was right... on the last lecture the room was pretty much empty :P –  Dunno May 15 at 19:30
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This is turning into the same thing as this Math Overflow question –  Ryan Reich May 15 at 22:15

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In a cup of coffee, one molecule of coffee is in its original location, even though the contents are undergoing convection. This is the Brouwer fixed point theorem.

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Also related I believe, a point on a map will correspond to the geographical point that you are standing at. –  illysial May 15 at 18:54
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Is that even true though? Molecules are pretty big. –  user973810 May 15 at 19:28
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@user973810 Not in precisely the same position(/orientation), but you still have at least one overlapping its original position (in non-quantum fashion) for finite-sized entities (assuming zero space between). But then, the coffee cup is just a donut anyway. –  Steven Stadnicki May 16 at 3:30
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Is this actually true? I can imagine a trivial 2 molecule coffee where this wouldn't occur (swap them). Looking at the theory itself it seems to include the condition that "assuming that the final position of each point is a continuous function of its original position" which may be the get-out clause. I can certainly see how this would work for a (non torn up) piece of paper because that is a continous function but I'm less convinced by a coffee –  Richard Tingle May 16 at 14:16
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@RichardTingle makes a good point. This falls down if you start thinking about molecules individually. Basically the underlying idea is that a function which takes the starting position of each point in the coffee to its current position must have a fixed point. So you have to consider the fluid as a uniform medium and forget it's actually just a bunch of particles bouncing around with reckless abandon. –  Ben Millwood May 17 at 1:07

$\sqrt[n]{2}$ is not rational for $n\geq 3$

Proof: If $\sqrt[n]{2}=\frac{p}{q}$ then $q^n+q^n=p^n$ contradicting Fermat's last theorem.

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Sourced to W.H. Schultz here: mathoverflow.net/a/42519/22971 –  Benjamin Dickman May 16 at 6:09
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Wow. That's an excellently simple way to motivate why one might care about whether Fermat's last theorem is true or not. –  David H May 16 at 13:03
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@DavidH I'm not convinced... the proof that $\sqrt[n]{2}$ is irrational is roughly identical to that of the irrationality of $\sqrt 2$ (e.g., very simple.) –  Mike Miller May 16 at 15:35
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Is that a "real life" thing? –  Matemáticos Chibchas May 17 at 3:03
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This is a neat "trick", but off-topic in my opinion. –  Daniel R May 17 at 8:26

The $n=2$ case of the Borsuk-Ulam theorem can be visualized by saying there exists some pair of antipodal points on the Earth with equal temperatures and barometric pressures. Of course, this is assuming that temperature and pressure varies continuously.

Ramsey's theorem says that, if given a sufficiently large complete graph that has been arbitrarily colored with $n$ colors, then one can find a monochromatic complete subgraph of a particular size. One example is as follows: Given any $2$-coloring on $K_6$, then we are guaranteed to find a monochromatic subgraph of size $3$. This has an interesting real-life interpretation: If we invite $6$ people to a party, then at least $3$ of them must be mutual acquaintances, or at least $3$ of them must be mutual strangers.

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I shared this one recently, and I think it is pretty funny. Using cohomology, we can prove that the alternating sum of binomial coefficients vanishes: $$\sum_{j=0}^n {n \choose j} (-1)^j=0.$$

Let $X=(S_1)^n$ be the $n$-dimensional torus. By the Künneth formula, and by the cohomology of $S^1$, $H^j(X, \mathbf Q)$ has dimension ${n \choose j}$. Therefore, the Euler characteristic of $X$ is

$$\chi(X)=\sum_{j=0}^n (-1)^j \mathrm{dim}_{\mathbf Q}H^j(X, \mathbf Q) = \sum_{j=0}^n {n \choose j} (-1)^j.$$

On the other hand, $X$ is a compact Lie group; let $\sigma$ be an infinitesimal translation $X \to X$. By the Lefschetz fixed point theorem, $\chi(X)$ is equal to the number of fixed points of $\sigma$, i.e., $0$.

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See also –  user3490 May 16 at 6:40
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I love this proof! –  SDevalapurkar May 16 at 23:28
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I totally use $n$-dimensional tori outside of math class... –  recursive recursion May 18 at 0:39
    
For a simple proof of the first equation, note that $\sum_{j=0}^n\binomial nj (-1)^j$ is just the multinomial expansion of $(x+1)^n$ evaluated at $x=-1$. –  Christopher Creutzig May 18 at 20:21

There is the futurama theorem It is about a mind-swapping machine. The question solved is:

When is it possible to return all the minds to their original bodies?

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Whether this actually counts as a "real-life" example is debatable, but +1 for Futurama anyway. –  iamnotmaynard May 16 at 17:40

Theorem : You can't cut a square pizza into equal-sized triangles among an odd number of persons, each of whom receive one slice.

This is a application of Sperner's lemma and the 2-adic valuation. Here is a nice proof.

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You could cut it into four triangles where their areas equal 2x, 2x, x, and x. One person gets the two small triangles. That is, if there were three people to split the pizza among. –  Azmisov May 15 at 23:42
    
Your theorem is false. Demonstration: You can cut it into 2n triangles and give each person 2 slices. –  Hellion May 16 at 3:45
    
Ok, I have forgot to mention that each person take one slice. Or a better way to rephrase it : 'A square pizza can't be split into an odd number of triangles slices with same area'. –  user10676 May 16 at 12:40
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"square pizza", what an atrocious concept in itself. good thing math is against it! (+1) –  G. Bach Jul 25 at 0:19

This is a fairly simple, but interesting application.

The mean-value theorem is used in some Automatic number plate recognition systems, based on average speed measurement. Using Optical character recognition, a computer system reads car plates at several fixed points along a road, and then uses the distance between these points to compute the car's average speed. By the mean-value theorem, the car must have gone at precisely that speed somewhere between the fixed points. A penalty can be issued accordingly, if this average speed exceeds the speed limit.

The system is used in several countries. It is not allowed in California, via Section 40801, Speed Trap Prohibition, of the California Vehicle Code, that reads

No peace officer or other person shall use a speed trap in arresting, or participating or assisting in the arrest of, any person for any alleged violation of this code nor shall any speed trap be used in securing evidence as to the speed of any vehicle for the purpose of an arrest or prosecution under this code.

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My calculus instructor used this very application to introduce the mean value theorem. –  MJD May 17 at 18:10
    
"A penalty can be issued accordingly, if this average speed exceeds the speed limit" ... and given that speeding is a strict liability offence. Otherwise you'd need to investigate the circumstances at the point at which the illegal speed occurred, which is difficult given only an existence proof ;-) –  Steve Jessop May 18 at 1:37
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I’m sure some drivers would call that a mean application of some value theorem. –  Christopher Creutzig May 18 at 20:26
    
You don't really need the mean value theorem here. $\int_a^b\mathrm{d}t\: f(t) \leq (b-a)\cdot \max\{f(t) | a\leq t\leq b\}$ is sufficient to prove the penalty, which holds also for non-continuous functions. –  leftaroundabout May 18 at 22:27

I had a teacher that liked to invoke Fubini's theorem to interchange the order of two finite sums.

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Ramsey's theorem has already been mentioned, but there is a key anecdote that deserves to be better known, and it is a great way to motivate the area. This comes from The Princeton companion to mathematics, Timothy Gowers, June Barrow-Green, and Imre Leader, eds., PUP, 2008. Chapter IV.19, Extremal and Probabilistic Combinatorics, pp. 562-575, written by Noga Alon and Michael Krivelevich, starts with the following story:

In the course of an examination of friendship between children some fifty years ago, the Hungarian sociologist Sandor Szalai observed that among any group of about twenty children he checked he could always find four children any two of whom were friends, or else four children no two of whom were friends. Despite the temptation to try to draw sociological conclusions, Szalai realized that this might well be a mathematical phenomenon rather than sociological one. Indeed, a brief discussion with the mathematicians Erdős, Turán, and Sós convinced him this was the case.

What Szalai was observing, in the language of Ramsey theory, is that $r(4,4)\le 20$, that is, any graph on $20$ vertices either contains a copy of $K_4$, or else it contains an independent set of size $4$ (actually, $r(4,4)=18$).

Alexander Bogomolny's website Cut the knot also recounts this story, and contains a proof of the equality $r(4,4)=18$, that has also been asked here a couple of times.

Another paper that recounts the story is Ramsey theory and the history of pre-Christian England, by William Gasarch. This paper goes on to present another application of this result. Unfortunately, as far as I can tell, the new story there appears to be fictional.

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You cannot create a "good" (continuous) function that for each 3D direction (unit vector) gives you an orthogonal direction.

Seriously. You cannot construct a "good" perpendicular to a vector.

Why? Blame the Hairy ball theorem. If you could construct such function $f(\vec v) : \forall \vec v f(\vec v) \perp \vec v $, then you have essentially constructed a continuous tangent field for a sphere $\vec v \to \{\vec v \times f(\vec v)\}$ which is impossible for dimensions greater than 2.

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I blame the Hairy Ball Theorem for pretty much everything. –  Alex May 16 at 15:11

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem". Have you more examples which are particular interesting or instructive?

This may be neither interesting nor instructive, but the canonical example I have repeatedly heard of this (which was once posed to me I believe on a second year calculus final exam) is: at time zero a turkey at temperature whatever is placed some distance above a heating element at temperature whatever. Assume the heating element is an infinite flat plane and the turkey is a uniform sphere of radius whatever, and that the thermal conductivity of turkey is whatever; a spherical turkey is safe to eat when its internal temperature is everywhere greater than whatever. What is the earliest time we can safely eat the turkey?"

This is of course a variation on this joke: http://en.wikipedia.org/wiki/Spherical_cow. More on the subject can be found here: http://physics.about.com/od/physicsintherealworld/p/TurkeyPhysics.htm

In the intervening two decades I have sadly forgotten how to solve the Heat Equation. I have roasted a great many (nonspherical!) turkeys since then; I use a thermometer to tell me when its done.

I also had a professor for first year calculus who liked to use chickens to illustrate theorems. A typical moment from a lecture: "If you have an infinite plane containing an infinite number of chickens, and you have infinitely many chickens inside a pen, and finitely many chickens outside the pen (waves arms wildly to illustrate infinitely many chickens in a bounded area) you just make your arms bigger (waves even more wildly) and then you have the entire set of chickens in the pen." Which was much more memorable than "the union of an infinite but bounded set with a finite set is a bounded set."

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This actually is (almost) real-life enough. Though shouldn't we also assume the spherical turkey above an infinite planar heating element to be in a vacuum? –  iamnotmaynard May 16 at 17:45
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@iamnotmaynard: Indeed. Also of course we must assume that there is no heat energy consumed due to the latent heat of evaporation, of the moisture within the turkeysphere. –  Eric Lippert May 16 at 18:03
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@iamnotmaynard: However we can then slice the heated turkey up into finitely many parts and reassemble it into a larger turkey, which is handy when extra people show up for dinner. –  Eric Lippert May 16 at 18:09

If you get up at 7:24 AM, you miss the bus. If you get up at 6:56 AM then you have to wait too long. After many tries, you find that if you get up at 7:09 AM you arrive to the bus stop at the same time as the bus, so you sleep to the maximum but you can still get the bus. That's Bolzano's Theorem for you.

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I'm not sure if the various puzzles about the non-planarity of $K_{3,3}$ count as real-life. Or in fact several other things from graph theory such as the Königsberg bridge problem, or the marriage problem (where I doubt that any community "optimizes" marriage by studying a bipartite graph).

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Olga Taussky-Todd and John Francis are two of the most recognizable figures in matrix analysis. Both were directly involved with the study of flutter (dynamic instability) of aircraft wings, while working in London. Taussky worked at the National Physical Laboratory, during World War II, and Francis worked at the National Defense Research Corporation, in the late 50s.

Studying flutter leads to delicate boundary problems in partial differential equations, but R.A. Frazer recognized its close connection with the problem of locating the eigenvalues of a matrix.

In abstract, a useful theoretical tool to attack this problem is Geršgorin's theorem, though its efficiency may vary wildly from matrix to matrix: Suppose $A$ is an $n\times n$ matrix with complex entries $a_{ij}$, $1\le i,j\le n$. If, for each $i$, we let $r_i=\sum_{j=1,j\ne i}^n|a_{ij}|$, then for each eigenvalue $\lambda$ of $A$ there is an $i$ such that $|\lambda-a_{ii}|\le r_i$. That is, the eigenvalues of $A$ are located in the union of the $n$ closed discs centered at the diagonal entries of $A$, $\{z\in\mathbb C\mid |z-a_{ii}|\le r_i\}$, $1\le i\le n$.

Taussky writes:

A large group of young girls, drafted into war work, did the calculation on hand-operated machines, following the instructions of Frazer and his assistants.

As described in section 6 of How I became a torchbearer for matrix theory, Geršgorin's theorem proved indeed to be a key tool in carrying out the relevant computations.

Once again, I didn't ask to be assigned to matrix problems. They found me.

By the time Francis came to work on the problem, the connection with computation of eigenvalues was well established, and Francis was assigned to write the relevant computer programs to carry out this task. Trying to accelerate the time the computations required, he created the shifted $QR$-algorithm which, even nowadays, is one of the most efficient tools to compute eigenvalues (and, via companion matrices, roots of arbitrary polynomials).

As David Watkins writes in Francis's algorithm:

[Francis's method] became and has continued to be the big workhorse of eigensystem computations. A version of Francis’s algorithm was used by MATLAB when I asked it to compute the eigensystem of a $1000\times 1000$ matrix on my laptop.

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Another pizza based example:

Take a slice of pizza. To keep the tip from flopping down, you pinch it between your fingers so that it gets a crease in radial direction. Works if the pizza is not too soggy.

Explanation: Theorema Egregium. Assuming the pizza is originally a rigid 2D surface with Gaussian curvature 0, it can only bend in one direction so that the product of the principal curvatures stays 0. By forcing the "harmless" curvature to be != 0, the "bad" curvature must stay 0.

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Euler's Theorum forms the basis of the RSA encryption system that's used in virtually every form of web-based cryptography (most notably, SSL and TLS protocols). "A Practical Application For Prime Numbers" provides an explanation better than I ever could. Regardless, the practical implications of this theorem are vast, and widely used (even right now, as I am about to submit this reply).

/I'm not a mathematician, but felt compelled to respond.

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