An example of a beautiful proof that would be accessible at the high school level?

Question

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about beauty in the teaching of school mathematics.

I'm trying to collect examples of good, accessible proofs that could be used in middle school or high school. Here are two that I have come across thus far:

(1) Pick's Theorem: The area, $A,$ of a lattice polygon, with boundary points $B$ and interior points $I$ is $A = I + B/2 - 1.$

I'm actually not so interested in verifying the theorem (sometimes given as a middle school task) but in actually proving it. There are a few nice proofs floating around, like one given in "Proofs from the Book" which uses a clever application of Euler's formula. A very different, but also clever proof, which Bjorn Poonen was kind enough to show to me, uses a double counting of angle measures, around each vertex and also around the boundary. Both of these proofs involve math that doesn't go much beyond the high school level, and they feel like real mathematics.

(2) Menelaus Theorem: If a line meets the sides $BC,$ $CA,$ and $AB$ of a triangle in the points $D, E,$ and $F$ then $(AE/EC) (CD/DB) (BF/FA) = 1.$ (converse also true) See: http://www.cut-the-knot.org/Generalization/Menelaus.shtml, also for the related Ceva's Theorem.

Again, I'm not interested in the proof for verification purposes, but for a beautiful, enlightening proof. I came across such a proof by Grünbaum and Shepard in Mathematics Magazine. They use what they call the Area Principle, which compares the areas of triangles that share the same base (I would like to insert a figure here, but I do not know how. -- given triangles $ABC$ and $DBC$ and the point $P$ that lies at the intersection of $AD$ and $BC,$ $AP/PD = \operatorname{Area} (ABC) / \operatorname{Area}(DBC).$) This principle is great-- with it, you can knock out Menelaus, Ceva's, and a similar theorem involving pentagons. And it is not hard-- I think that an average high school student could follow it; and a clever student might be able to discover this principle themselves.

Anyway, I'd be grateful for any more examples like these. I'd also be interested in people's judgements about what makes these proofs beautiful (if indeed they are-- is there a difference between a beautiful proof and a clever one?) but I don't know if that kind of discussion is appropriate for this forum.

Edit: I just want to be clear that in my question I'm really asking about proofs you'd consider to be beautiful, not just ones that are neat or accessible at the high school level. (not that the distinction is always so easy to make...)

Answer 1

Extending on Ralph's answer, there is a similar very neat proof for the formula for $Q_n:=1^2+2^2+\dots+n^2$. Write down numbers in an equilateral triangle as follows:

    1
   2 2    
  3 3 3
 4 4 4 4

Now, clearly the sum of the numbers in the triangle is $Q_n$. On the other hand, if you superimpose three such triangles rotated by $120^\circ$ each, then the sum of the numbers in each position equals $2n+1$. Therefore, you can double-count $3Q_n=\frac{n(n+1)}{2}(2n+1)$. $\square$

(I first heard this proof from János Pataki).

How to prove formally that all positions sum to $2n+1$? Easy induction ("moving down-left or down-right from the topmost number does not alter the sum, since one of the three summand increases and one decreases"). This is a discrete analogue of the Euclidean geometry theorem "given a point $P$ in an equilateral triangle $ABC$, the sum of its three distances from the sides is constant" (proof: sum the areas of $APB,BPC,CPA$), which you can mention as well.

How to generalize to sum of cubes? Same trick on a tetrahedron. EDIT: there's some way to generalize it to higher dimensions, but unfortunately it's more complicated than this. See the comments below.

If you wish to tell them something about "what is the fourth dimension (for a mathematician)", this is an excellent start.

Answer 2

The theorem of "friends and strangers": the Ramsey number $R(3,3)=6$. Not only can the proof be understood by high-school students, a proof can be discovered by students at that level via something akin to the Socratic method. First students can establish the bound $R(3,3) > 5$ by 2-coloring the edges of $K_5$:
          
Then they can reason through that a 2-coloring of the edges of $K_6$ must contain a monochromatic triangle, and so $R(3,3)=6$: in every group of six, three must be friends or three must be strangers.

After this exercise, an inductive proof of the 2-color version of Ramsey's theorem is in reach.

An added bonus here is that one quickly reaches the frontiers of mathematics: $R(5,5)$ is unknown! It can be a revelation to students that there is a frontier of mathematics. And then one can tell the Erdős story about $R(6,6)$, as recounted here. :-)

Answer 3

Euler's Bridges of Konigsberg problem. You can give it to students for five minutes to play with, watch them get annoyed, and then offer them the classical simple and beautiful impossibility proof. I think a lot of high school students, and even bright middle school students, would be totally convinced.

Answer 4

The proof, by counting inversions, that you can't interchange the 14 and 15 in the 15 puzzle, just by sliding, is accessible to high-school students, introduces important ideas, and might be found beautiful.

Answer 5

The trefoil knot is non-trivial.

Proof: It has a tricoloring:

^(source)

And the existence of a tricoloring is preserved by Reidermeister moves. QED

Answer 6

Two of my favorites: (1) The parameterization of all primitive Pythagorean triples; (2) The formula for the $n$th Fibonacci number in terms of the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$, with the corollary that $\displaystyle \lim_{n \longrightarrow \infty} F_n/F_{n-1} = \phi$.

Answer 7

Going by the parameters of the question, I don't see why the proof would necessarily need to be of a sophisticated theorem. I think Euclid's proof of the infinitude of primes is beautiful and definitely accessible to a high school audience. Having given the proof, one might reflect on some of its features that generalize to many other contexts, like proof by contradiction or the ability to use a clever construction to avoid infinite enumeration.

Answer 8

Cantor's diagonal argument.

The warm-up could be an equally beautiful proof, namely that the rationals are countable.

Answer 9

Seeing the struggle of many students with standard trigonometry, I especially like the rational parametrization of $x^2+y^2=1$ (which is equivalent to listing all Pythagorean triples) by starting from $\sin^2\phi+\cos^2\phi=1$ and then using $$ \sin\phi=\frac{2t}{1+t^2}, \quad \cos\phi=\frac{1-t^2}{1+t^2}, \qquad t=\tan\frac{\phi}2. $$ Note that the formulas are usually used in the context of integration of rational expressions in sine and cosine.

At the same time, a more general "geometric" argument (applicable to general quadratics), due to Bachet (1620), is still at school level. Namely, fix a single rational point on the curve, $(x _ 0,y _ 0)$ say, and consider the intersection points of the curve and straight lines $y-y_0=t(x-x_0)$ with rational slope $t$ passing through the point.

A beauty here is because of variety of different geometric and analytic methods for solving a classical arithmetic problem.

Answer 10

The Gale-Sharpley stable marriage theorem, http://en.wikipedia.org/wiki/Stable_marriage_problem .

The algorithm and its proof are very much accessible to school students. Despite its innocuous look, the algorithm is not easy at all to invent.

On a similar note, Hall's theorem: http://en.wikipedia.org/wiki/Hall%27s_marriage_theorem#Graph_theory . This looks like a recreational puzzle but actually is closer to university mathematics than everything done in high school.

Here is another combinatorial exercise which, properly presented, does not even look like mathematics: the round track puzzle. The thing I don't like about it is that the standard "gotcha" proof (explained in the usual, informal way) requires a bit too much concentration to understand - some students might fail at it and take it as an example that mathematical proofs are something one either believes or not, rather than something one can check. Of course, one can formalize the proof, but this requires quite an amount of time in a high school class. [Update, a decade later: I have written an expanded version of the solution up as the solution to Exercise 5.2.4 in my Fall 2020 Math 235 notes.]

Answer 11

The proof that $\sqrt{2}$ is irrational is a nice example of proof by contradiction.

Answer 12

The Halting Problem. The first time I saw this was my senior year in high school and it completely blew me away. All you need is a notion of what an algorithm is and very basic logic (enough to recognize that assuming $A$ and deriving $\neg A$ is a contradiction)

In a similar vein, Russell's Paradox. The problem here is that you need some basic set theory, so this is more for advanced high school students.

The first beautiful proof I saw in high school (it was beautiful at the time, but now seems too trivial) was the fact that for a geometric series $a, ax, ax^2, \dots$ the sum of the first $n$ terms is $a \cdot \frac{1-x^n}{1-x}$. I thought this was cool because of all the cancellation that seems to come out of nowhere. The treatment here is nice.

Answer 13

This topological proof of the fundamental theorem of algebra is accessible to high school students, particularly those at the precalculus level.

     ^(source)

There are two major problems with this. First, while the winding number is intuitive, it takes effort to define it rigorously. Second, you also want to establish the basic property that the winding number doesn't change as you deform a curve without going over the origin, which again is difficult to establish rigorously without topology. Without these details, you might call this a hand-waving argument instead of a proof. It's good to give references to where these results will be established rigorously, and to give arguments for other results which are more complete.

Nevertheless, I like presenting this proof for several reasons. I think it's beautiful. Geometrically, what $x\mapsto x^n$ does to the complex plane is easy to understand, but many students have little intuition about what this map does, only what polynomials look like on the real line. So, this argument doesn't just say that the statement is true, it is illuminating. The fundamental theorem of algebra is also a result students encountered in algebra, but they usually don't know why it's called a theorem. This is also an opportunity to talk a little about what is studied in more advanced areas of mathematics. It can lead into discussions of topology or the difficulty solving polynomials by radicals.

Answer 14

One of the keys to making a proof accessible to high school students (or just non-mathematicians) is to make the answer relevant. This gives a dual responsibility, to ensure that the theorem is motivated and that the proof is accessible. The proof of the infinity of the primes has been mentioned already and is a fantastic example. You can lead students in to it using the (almost trivial) proof that there is no largest integer.

Another example is the classification of the regular polyhedra. With good students and models you can even lead them to the proof there there are at most 6 regular polytopes in 4d (actually showing they all exist is a little harder).

Keeping with polyhedra, the Euler characteristic is also powerful. Start with balloons and get the students to draw lines freely so you get a tiling. Then get them to count faces, vertices and edges. David Eppstein collected 19 proofs to choose from, several of which would be perfect for non-mathematicians: http://www.ics.uci.edu/~eppstein/junkyard/euler/

As a final example (and to show that it does not have to be deep mathematics to motivate) you can consider the question of blocking a square on a chess board and filling the remainder with tromioes. It starts with a puzzle, you can get people to play with, and leads to a lovely induction proof: http://www.cut-the-knot.org/Curriculum/Games/TriggTromino.shtml

Actually polyominoes are a fantastic source of many other fun, non-trivial but accessible proofs.

Answer 15

I've been collecting simple, often one-step, proofs.

http://www.cut-the-knot.org/proofs/index.shtml

Some I judge beautiful - these are listed separately.

Answer 16

I recommend Kelly's proof of the Sylvester-Gallai theorem (the original proof of Gallai was about 30 pages long, this one takes a few lines). The theorem and the proof can be read here.

Answer 17

Im in high school and i loved the proof of the fermat-toriccelli point of a triangle.

Answer 18

Minkowski's Theorem (every convex region in the plane of area greater than 4 that's symmetric about the origin contains a lattice point other than (0,0)) is not at all obvious (are you sure you can't squeeze a sufficiently large "blob of irrational slope" in there?) but has a beautiful, simple, and surprising geometric proof.

Answer 19

For someone in high school, I think it's good to prove that the sum of the interior angles of a triangle is $\pi$ if they don't know why. Personally, I was never shown why this fact is true, and I feel that it's generally a bad idea to not know why something in math is true, especially when the answer is pretty. My favorite proof is to think about how the normal vector changes as you walk around the triangle -- it's nice because it generalizes to other shapes (which may not even be polygons).

Answer 20

The formula $1 + 2 + \cdots + n = n(n+1)/2$ can be proved on middle school level: Assume first $n$ is even. Then there are $n/2$ pairs $(1,n), (2,n-1), \ldots, (n/2,n/2+1)$ those sum is always $n+1.$ Thus the overall sum is $n/2\cdot(n+1).$ The case when $n$ is odd can be treated in the same manner.

Answer 21

1) Many elementary binomial identities or identities with Fibonacci numbers have beautiful proofs. Let me only mention the matrix representation of Fibonacci numbers whose determinant gives Cassini's identity.

2) Another elementary problem is the following: Is it possible to cover a checkerboard from which one white and one black square have been removed with dominoes? To show that this is possible run through the board in a cyclical way. Observe that on this path between a white and a black square are an even number of squares. Since I don't know how to make figures I indicate such a path for a 4x4-board: ((1,1),(1,2),(1,3),(1,4),(2,4),(2,3),(2,2),(3,2),(3,3),(3,4),(4,4),(4,3),(4,2),(4,1),(3,1),(2,1)).

Answer 22

You should certainly look at the two books by Ross Honsberger, "Mathematical Gems I" and "Mathematical Gems II". A favourite example of mine the proof due to Conway that there are configurations of checkers below the half plane on an infinite board that allow you to move a checker 4 rows into the upper half plane, but not five rows.

The negative result is an ingenious argument using nothing more than the quadratic formula, but provides a great example of to apply mathematics in unexpected contexts.

Answer 23

I suggest Euler's polyeder formula with application to the Platonic solids. One first observes that instead of polyeders one can consider graphs in the plane, counting the unbounded region as a face. One observes next that one can just as well allow the edges to be pieces of curves. Then one observes that the formula $F - E + V = 2$ is preserved if one edge or one vertex is added, thus the formula is proved by induction. Then use the formula to prove that there are no regular polyeders except the five Platonic solids as follows: faces can only be triangles, squares or pentagons, edges are always common to exactly two faces, and for the number of faces that meet in a vertex there are a number of possibilities that one checks. For each case, $E$ and $V$ can be expressed in terms of $F$, and the formula gives the possible values of $F$.

Answer 24

i suggest the proof archimedes wanted on his tombstone and its relatives. i.e. since two solids with the same horizontal slice area at every height have the same volume, hence by pythagoras, the volume of a cylinder equals the sum of the volumes of an inscribed cone and an inscribed solid hemisphere.

to generalize this, the volume generated by revolving a solid hemisphere around a planar axis in 4 space equal that generated by revolving a cylinder minus that generated by revolving a cone. Using the fact that the center of gravity of a cone is 1/4 the way up from the base, one obtains the volume of a 4-sphere, as pi^2/2 R^4.

a generalization of the first computation is that of the volume of a bicylinder (intersection of two perpendicular cylinders), since it is the difference of the volumes of a cube containing the bicylinder and a square based double pyramid also inscribed in the cube.

I find these beautiful, but of course that is subjective.

I also like euclid's argument for pythagoras, and for constructing a regular pentagon, but they are hard to reproduce here briefly.

Answer 25

Here is one that I like and used it for different purposes, e.g. introduction to proofs, algebraic thinking, beauty, and so one. Shuffle a deck of cards. Divide it into two halves. Magic: The number of the red cards in one of the halves is exactly equal to the number of black cards in the other half. It has a simple algebraic proof. However, The bigger magic is yet to come.

Theorem: Vertically opposite angles are equal.

Bigger Magic: The two theorems/proofs are essentially the same!

Answer 26

There is a very elegant proof that there exists no continuous injection from the plane into the real line. The proof can basically be given by drawing a picture on the blackboard.

Suppose there is such an injection $f$. Let $x$ and $y$ be distinct points in the plane and let $g_1$ and $g_2$ be paths from $x$ to $y$ such that $g_1(r_1)\neq g_2(r_2)$ for $r_1,r_2\in (0,1)$. Now this implies that $f\circ g_1((0,1))\cap f\circ g_2((0,1))=\emptyset$, contradicting the intermediate value theorem.

Answer 27

Morley's Theorem.

Wikipedia's proof is completely elementary and only involves trigonometric identities and euclidean plane geometry.

There is also a proof by Alain Connes, based on affine geometry techniques. Of course it is a bit more technical, but again it involves math that doesn't go much beyond the high school level, and could be appreciated by the most gifted students

Answer 28

Those are pretty nice, and can be done at a fairly low level (say, from 12 years old onwards) :

• The proof of the formula "half base times height" for the area of a triangle, by first considering a right triangle and completing a rectangle, then considering an arbitrary triangle and breaking it in two along an height (two cases : inside(+) or outside(-)) : it is a nice example of how mathematicians treat the general case by reduction to particular cases ;
• Euclid's proof of the pythagorean theorem using the previous formula, as in this animation.

Answer 29

Using Euler's formula ($F-E+V=2$, as mentioned earlier), it can be proved that the graphs $K_5$ and $K_{3,3}$ are not planar. I think these proof and also the proof of Euler's formula are simple enough to be understood by an interested high school student.

Answer 30

In the general game "Poset Chomp" the first player always has a winning strategy. The proof is a one-line strategy stealing argument, hence nonconstructive. In fact, a winning strategy is unknown in most cases, which makes the result interesting and mysterious. For a good quick account see here.