Ask Uncle Colin: how big do the patches on a football need to be? | Colin

Dear Uncle Colin,

I’m trying to sew a traditional football in the form of a truncated icosahedron. If I want a radius of 15cm, how big do the polygons need to be?

— Plugging In Euler Characteristic’s Excessive

Hello, PIECE, and thank you for your message!

Getting an exact answer to that is a little tricky, but we can come up with a pretty good approximation: if we assume the ball’s surface area is the same as that of a sphere, we can work it out.
The truncated icosahedron

Now, a truncated icosahedron is made of 12 regular pentagons and 20 regular hexagons, all of the same side…

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The Mathematical Ninja and the Poisson Distribution | Colin

“What are the ch…”

“About 11.7%,” said the Mathematical Ninja. “Assuming $X$ is drawn from a Poisson distribution with a mean of 9 and we want the probability that $X=7$.”

“That’s a fair assumption, sensei,” pointed out the student, “given that that’s what the sodding question says.” A wiser student may not have pushed their luck. “So, where does 12% come from?”

“It’s simple,” said the Mathematical Ninja. “You know, of course, that $P(X=n) = e^{-\lambda} \frac{\lambda^n}{n!}$?”

“But of course! I remember it because it’s the $n$th term of a Maclaurin series for $e^{-\lambda} e^{x}$…

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Ask Uncle Colin: Parametric Second Derivatives | Colin

Dear Uncle Colin,

I have a pair of parametric equations giving $x$ and $y$ each as a function of $t$. I’m happy with the first derivative being $\diff{y}{t} \div \diff{x}{t}$, but I struggle to find the second derivative. How would I do that?

– Can’t Handle An Infinitesimal Nuance

Hi, CHAIN, and thanks for your message!

This always used to trip me up as well – it stood to reason that if the first derivative was $\diff yt \div \diff xt$, then the second derivative should be $\diffn 2yt \div \diffn 2xt$. If only life were so simple.

Instead, the thing to do is to treat your first…

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Ask Uncle Colin: A Short, Sweet Limit | Colin

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

What is $\lim_{x \to \infty} \left\{ \sqrt{x^2 + 3x} – x\right\}$?

– Raging Over Obnoxious Terseness

Hi, ROOT, and thanks for your very brief question.

My approach would be to split up the square root and use either a binomial expansion or completing the square, as follows:

$\sqrt{x^2 + 3x} = x\sqrt{1 + \frac{3}{x}}$….

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Wrong, But Useful: Episode 43 | Colin

In this month’s installment of Wrong, But Useful, our special guest co-host is @mathsjem (Jo Morgan in real life) from the indispensable resourceaholic.com.
We start by talking about resourceaholic.com and how Jo manages to fit such a punishing blog schedule around being a nearly-full-time maths teacher.
Colin wonders how writing has affected Jo’s teaching practice.
The number of the podcast is 530, an untouchable number.
Apparently you peasants who carry money around are excited about the new £1 coin. Jo used to work at the Bank of England and has insights about how statistics can be…

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The Mathematical Pirate and The Quotient Rule | Colin

“Arr, that be a scurvy-lookin’ expression!” said the Mathematical Pirate. “A quartic on the top and a quadratic on the bottom. That Ninja would probably try to factorise and do it all elegant-like.”

“Is that not the point?”

“When you got something as ‘orrible as that, it’s like puttin’ lipstick on a shark. There ain’t all that much point, and it’s just gonna annoy the shark.”

The student looked both ways. It was not a metaphor he was familiar with. “Right you are, then, captain. So we’ve got to find the gradient of $f(x) = \frac{x^4 + x^3 – 13x^2 + 26x – 17}{x^2 – 3x + 3}$ when $x=1$….

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Ask Uncle Colin: Approximating an embedded exponential | Colin

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

Help! My calculator is broken and I need to solve – or at least approximate – $0.1 = \frac{x}{e^x – 1}$! How would you do it?

— Every $x$ Produces Outrageous Numbers, Exploring New Techniques

Hi, ExPONENT, and thanks for your message!

That’s a bit of a beast, and you’ll not get an exact solution using elementary…

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