I had a fascinating conversation on Twitter the other day about, I suppose, different modes of solving a problem. Here’s where it started:

Heh. You spend half an hour knee-deep in STEP algebra, solve it, then realise that tweaking the diagram a tiny bit turns it into a two-liner.

— Colin Beveridge (@icecolbeveridge) February 5, 2019

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I intended it as a throwaway comment, but it got some interesting responses.

@colinthemathmo (Colin Wright) pointed out:

It’s an interesting conundrum … Not all problems have neat solutions, so we have to have the full power of algebra at our disposal. Should we then still look for the elegant insight, or simply crack on and just do it?

— Colin Wright (@ColinTheMathmo) February 5, 2019

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… which, as Colin’s messages are wont to do, gets straight to the heart of things: is maths about getting the answer, or about getting the insight?

Of course, it’s both – but it’s a rare answer that makes me go ‘oo!’

@ajk44 (Alison Kiddle) suffers from the same difficulty as I often do:

Quite often, my brain CAN’T make sense of an elegant insight unless I’ve gone the long way round first. Or perhaps that’s just the excuse I make for always going the long way round…

— Alison Kiddle (@ajk_44) February 5, 2019

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@DavidB52s (David Bedford) took it on from a more pedagogical point of view:

I always try to present direct arguments to students first, no matter how messy (the arguments not the students). The proof that permutations are either even or odd is a good example. The alternating polynomial proof is elegant but I still want to examine cases directly.

— David Bedford (@DavidB52s) February 5, 2019

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And @RobJLow (Robert Low) followed up with this:

Took me a while to realize that I couldn’t just give them my hard-won understanding by explaining how I think of it. It doesn’t work like that. But I do like to think I’ve got a bit better at providing a first version that lays the foundation for a deeper understanding…

— Robert Low (@RobJLow) February 5, 2019

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As the man said, you shouldn’t let other people get your kicks for you. Having someone else’s epiphanies for them is just impossible.

So what are your thoughts about epiphanies and elegance? Is there a good way to teach them? I’d love to hear your comments.

## Barney Maunder-Taylor

To illustrate: “a stick is snapped randomly into three pieces. What is the probability that those three pieces could be rearranged to make a triangle?”. This puzzle took me two days to solve, and only after getting two incorrect answers, some integration, and performing a simulation to check which of my answers was correct. Two pals then independently came up with the correct answer after barely more than a moment’s thought and no integration. What’s the point? Not sure to be honest, except that I did have enormous fun working through the problem and eventually cracking it successfuly – by whatever means, and the fact that I’d gone a needlessly long way has, peversely enough, only added to my enjoyment of it.

## Colin

That’s exactly it – I find that realising there’s a quick way *after* doing it the long way makes me appreciate the quick way more.