Posted in ask uncle colin.

Dear Uncle Colin, How would I work out $\sqrt(\ln(100!))$ in my head? – Some Tricks I’d Really Like In Number Games Hi, STIRLING, and thanks for your message! I don’t know how you’d do it, but I know how the Mathematical Ninja would! Stirling’s Approximation1 says that $\ln(n!) \approx n

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Posted in algebra.

In class, a student asked to work through a question: Let $f(x) = \frac{5(x-1)}{(x+1)(x-4)} – \frac{3}{x-4}$. (a) Show that $f(x)$ can be written as $\frac{2}{x+1}$. (b)Hence find $f^{-1}(x)$, stating its domain. The answer they gave was outrageous1. Part (a) Part (a) was fine: combine it all into a single fraction

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Posted in ask uncle 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 –

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Posted in podcasts.

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

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Posted in puzzles.

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. —

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Posted in ask uncle 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

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Posted in dome.

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

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Posted in ask uncle 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

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Posted in matrices.

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? pic.twitter.com/vxIKeS4Lrq — Robert Low (@RobJLow) June 26, 2018

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Posted in ask uncle colin.

Dear Uncle Colin, I’m told that $2^a \equiv 9 \pmod{11}$. How do I find $a$? – Powers And Stuff, Calculating And Learning Hi, PASCAL, and thanks for your message! As so often, there are (at least) two reasonable ways to tackle this: a brute force way and an elegant way.

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