CTK Insights

The identity $\sqrt[3]{2 \pm \sqrt{5}} = \frac{1 \pm \sqrt{5}}{2},$ although simple, is rather surprising. It could be verified by raising the right-hand side to the power of 3:

$(\frac{1 \pm \sqrt{5}}{2})^{3} = \frac{1}{8} (1 \pm 3\sqrt{5} + 15 \pm 5\sqrt{5}) = 2 \pm \sqrt{5}.$

One way of accidently running into this identity is by means of Cardano's cubic formula applied to the third degree equation $x^{3}-2x-4=0$ The formula will produce terms that include cubic roots. On the other hand, the equation has an integer root $x = 2$ and, therefore, the remaining two roots satisfy a quadratic equation $x^{2}+2x+2=0$ with integer coefficients. Professor Tom Osler from Rowan University has observed

It is known that $\sqrt[3]{5}$ is not constructible. Yet $\sqrt[3]{2 + \sqrt{5}},$ which looks more complicated than $\sqrt[3]{5},$ is constructible because it can be expressed by the golden section. This is an unexpected result. It is surprises of this type that make it a joy to be a mathematician.

A most engaging remark, indeed.

More recently, I had a similar experience on a smaller scale. Say, consider a problem: Prove that the equation $x^{2}+y^{2}=6xy$ has no solutions in positive integers. A more or less common way of approaching a problem like that would be to apply Fermat's infinite descent. (The case where at least one of $x$ or $y$ is assumed to be even is easy. Assuming both of them odd leads to no obvious contradiction.) But here's a simpler approach. By adding to and subtracting $2xy$ from both sides gives $(x+y)^{2}=8xy$ and $(x-y)^{2}=4xy.$ Divide one by another and take the square root: $\frac{x+y}{|x-y|}=\sqrt{2}.$ This is of course impossible as the left-hand side is rational, while the right-hand side is not.

Martin Gardner had two collections of surprises in this spirit - aha! Insight and aha! Gotcha - that are also available in a single volume.

Not every child is born with a mindset fit to see and appreciate those instances of - real, albeit "small" - mathematics. Most kids do not naturaly come across such small surprises. However, there are plenty of elegant mathematics even on a most elementary level.

Years ago the Australian mathematician M. L. Urquhart discovered a theorem while working on "some fundamental concepts of the theory of special relativity." He dubbed it the most elementary theorem of Euclidean geometry since it only involves the concepts of straight line and distance. Now known as Urquhart's theorem it reads, If ABB' and AC'C are straight lines with BC and B'C' intersecting in D and if AB + BD = AC' + C'D, then AB' + B'D = AC + CD. This is not what I mean by an "elementary level."

There are problems whose solution requires little or no up-front knowledge. Socrates' doubling (or halving) a square is one example. I'll list a few more.

A pin has two ends: a pointed one and a blunt one. If three pins form a triangle then either all vertices look the same (pointed/blunt) or all look different (pointed/pointed, blunt/blunt, blunt/pointed.)

4-letter word ladders, one of Lewis Carroll's inventions. The task is to convert one 4-letter word into another by changing a letter at a time so as to have meaningful 4-letter words at every step of the process. Claim: some of the words in the chain is bound to have two vowels.

Take two very different 3D bodies, say, a ball and a cube or a potato and a kettle, etc. On each body in the pair draw curves with a marker. Is it always (or never) possible to draw identical curves on any two bodies? (Fro an answer, imagine the two bodies made of some amorphous material so that they can physically intersect, i.e. penetrate each other. Their intersection will be a curve on the surfaces of both.)

Assume you mastered the Toads and Frogs puzzle. Then it must be easy for you to solve its 2D analog. Is it?

I think that what is missing from school curricula and from course offerings intended for potential teachers is this kind of simple, almost unversally accessible mathematical morsels. I actually would like to solicit additional examples of simple mathematics. Not mathematical facts, but statements that require and admit a proof, a proof no longer than a step or two, based more on logic and every day experience than on knowledge of some axioms.

To start with, it is always possible to comment on this post. If the thing picks up, I promise to find a more suitable arrangement. Actually, why not to declare a competition for the Most elementray aha! moments. A winner will be determined monthly and announced at the end of every month. A monthly prize of $1 will be paid to the winner's paypal account.

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