Ask Uncle Colin: It’s Hip To Be Square | Colin

Dear Uncle Colin,

I’m struggling to make any headway with this: find all integers $n$ such that $5 \times 2^n + 1$ is square. Any ideas?

Lousy Expression Being Equalto Square Gives Undue Exasperation

Hi, LEBESGUE, and thanks for your message!

Every mathematician should have a Bag Of Tricks – things they look out for that occasionally make problem-solving easier. Think of it as a mathematical toolkit. For me, it’s things like Clever Regrouping, Fermat’s Little Theorem, modulo arithmetic and – more often than the others – the difference of two squares.

It’s not immediately obvious…

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Wrong, But Useful: Episode | Colin

In this month’s episode of Wrong, But Useful, we’re joined by @DrSmokyFurby and his handler, Belgin Seymenoglu.

Apologies for the poor audio quality on this call. Dave’s fault, obviously1 .

We discuss:
The Talkdust podcast
(via Adam Atkinson): Life insurance
Superpermutations: new record for n = 7 in the comments on a YouTube post
@pecnut and @mscroggs have a LaTeX package that puts hats on things
Desmos adding support for distributions and all sorts
@mdawesmdawes’s QUIBANS website
Dave has been playing the game The Mind
(Via @peterrowlett), 318,000 combinations of pringles

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On Epiphanies | Colin

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

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…

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Ask Uncle Colin: Computing $\sqrt{2}$ | Colin

Dear Uncle Colin,

If I didn’t have a calculator and wanted to know the decimal expansion of $\sqrt{2}$, how would I be best to go about it?

Roots As Decimals – Irrational Constant At Length

Hi, RADICAL, and thanks for your message!

There are several options for finding $\sqrt{2}$ as a decimal (although why you’d want to know it beyond about three places, I have no idea).

One is the ‘long division’ method that @colinthemathmo has explained in more detail than I would ever care to, so I’ll just send you there if you’re interested in that.

But there are two other methods I would…

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The Dictionary of Mathematical Eponymy: Banach’s Matchbox Problem | Colin

Stefan Banach was one of the early 20th century’s most important mathematicians – if you’re at all interested in popular maths, you’ll have heard of the Banach-Tarski paradox; if you’ve done any serious linera algebra, you’ll know about Banach spaces; if you’ve read Cracking Mathematics (available wherever good books are sold), you’ll recognise his name from the chapter on the Scottish Cafe.

So obviously, I’ve picked a much less serious example from the list of things named after him: Banach’s matchbox problem. So far as I know, the problem is not due directly to Banach: how I imagine it…

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Ask Uncle Colin: An Implicit problem | Colin

Dear Uncle Colin,

I have to find the points $A$ and $B$ on the curve $x^2 + y^2 – xy =84$ where the gradient of the tangent is $\frac{1}{3}$. I find four possible points, but the mark scheme only lists two. Where have I gone wrong?

I’ve Miscounted Points Like I Can’t Infer Tangents

Hi, IMPLICIT, and thanks for your message!

I think I have an idea of what’s gone wrong. Let me talk through the question.
Differentiating implicitly

If you differentiate implicitly, you find that $2x + 2y \dydx – y – x \dydx = 0$. You could simplify that, but there’s really no need: you know that…

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A Matrix Definition of a Line | Colin

Every so often, I see a tweet so marvellous I can’t believe it’s true. Then I bookmark it and forget about it for months, until I don’t know what to write next.

An example is @robjlow‘s message from June:

Aren’t determinants wonderful?

— Robert Low (@RobJLow) June 26, 2018

a) Isn’t that lovely? and b) Hang on, is it really true?
Let’s algebra

Working out the determinant using my favourite diagonal stripes method gives $x(y_1 – y_2) + y(x_2 – x_1) + (x_1 y_2 – x_2 y_1) = 0$.

Does that look plausible? Certainly it’s a straight line, and certainly…

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