What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius, and the proof is actually short (say, one page or so) and uses elementary mathematics only?
[Math] Past open problems with sudden and easy-to-understand solutions
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Related Solutions
There is the "Futurama Theorem" (based on a television show by the same name) that might be interesting to them. Dr. Farnsworth created a machine that swaps the minds of two individuals (i.e. your mind would be in my body and my mind in your body). Unfortunately, the body builds up resistance after a swap, which prevents swapping the minds back directly (i.e. if you and I just switched minds, we could not switch back without one of us first spending time in someone else's body).
The question is, given a collection of jumbled up minds and bodies, how can you get them all back to normal?
Your students will quickly realize that extra "clean" bodies (individuals who haven't swapped minds with anyone) are needed, since a collection of just two people with swapped minds could never get back to normal. It turns out that, no matter the size of the collection or the permutation of minds, two extra "clean" bodies are sufficient to reverse the permutation.
Here's the proof that appeared in the show, which uses nothing other than basic facts about permutations:
You can also read about it at Wikipedia and CutTheKnot.
Niccolò “Tartaglia” Fontana invented the first general method to find the roots of an arbitrary cubic equation (based on earlier work by himself and others on how to solve cubics of particular forms), but kept his method secret so as to preserve his advantage in problem-solving competitions with other mathematicians.
He divulged the secret to his student Gerolamo Cardano on condition that Cardano keep the secret, and there was a bitter dispute between the two when Cardano broke his promise.
Best Answer
The integral of $\sec x$ stumped mathematicians in the mid-seventeenth century for quite a while until, in a flash of insight, Isaac Barrow showed that the following can be done:
$$\int \sec x \,\mathrm{d}x= \int \frac{1}{\cos x} \, \mathrm{d}x=\int \frac{ \cos x}{\cos^2 x} \, \mathrm{d}x=\int \frac{ \cos x}{1-\sin^2 x} \, \mathrm{d}x.$$
Using $u$-substitution and letting $u=\sin x$, the integral transforms to
$$\int \frac{1}{1-u^2} \, \mathrm{d}u,$$
which is easily evaluated by partial fractions.