Posted in puzzles.

It’s always fun to tackle a puzzle from one of Cav’s posts - in this case, a Catriona Shearer puzzle; it looks like my solutions are completely unrelated to his, although in reality I tend to sneak the odd peek and take some inspiration1 I also liked that Cav shared

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Posted in ask uncle colin.

Dear Uncle Colin, How do I tell which is larger, $2^{-10^{-20}}$ or $1 - 2^{-10^{20}}$? - Unexpectedly Narrow Interval… Thank You! Hi, UNITY, and thanks for your message! As you’ve doubtless realised, both of those are “pretty much 1”. The question is, which is closer? As usual, there are several

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Posted in dome.

A second-in-a-row Dictionary of Mathematical Eponymy post about Boolean logic today – and another example of a Very Neat Diagram. What is a Randolph diagram? You’ve seen - at least, I hope you’ve seen - Venn diagrams. Beastly things. I would chuck them out the window if I could, they

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Posted in ask uncle colin.

Dear Uncle Colin, Can two polynomials be equal everywhere in some non-trivial interval $[a,b]$ but not equal elsewhere? - Lacking A Good Reasonable Answer, No Good Explanation Hi, LAGRANGE, and thanks for your message! The answer is “no”, but to explain why, I need to give you a few pieces

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Posted in puzzles.

A puzzle that came to me via @sheena2907: Choose two numbers, $x$ and $y$, uniformly from $[0,1]^2$. What’s the probability that $\frac{x}{y}$ rounds to an even number? What’s the probability that it rounds down to an even number? As always, spoilers below the line. Rounding One of my best approaches

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Posted in ask uncle colin.

Dear Uncle Colin, Is there a rigorous explanation of why the direction of the inequality changes when you flip the sign? - Fuzzy Logic, Inequality Puzzle Hi, FLIP, and thanks for your message! Here are my best efforts at a rigorous explanation; I’d be interested to read anyone else’s go

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Posted in puzzles.

A nice thinker from Futility Closet: A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make? Feel free to

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Posted in podcasts.

In this, the penultimate episode of Wrong, But Useful, we’re joined by @karenshancock (who is Karen Hancock in real life). We discuss: A scoring system for an online quiz round. The Big Mathoff What is a surd? Currently reading Ratio by Michael Ruhlman - a cookery book that starts from

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Posted in ask uncle colin.

Dear Uncle Colin, I’m trying to solve $\frac{x}{x-1} = \frac{1}{x-1}$. I think the answer should be 1, but my teacher disagrees. What do you think? - First Results Are Contradicting Teachers’ - Is One Nonsense? Hi, FRACTION, and thanks for your message! It’s tempting, here, to multiply both sides by

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Posted in arithmetic, haskell.

This is an extended version of my entry in the Lockdown Mathoff at the Aperiodical Binet’s formula1 is a lovely way to generate the $n$th Fibonacci number, $F_n$. If $\phi = \frac{1}{2}\left(\sqrt{5} + 1\right)$, then $$F_n = \frac{ \phi^n - (-\phi)^{-n} }{\sqrt{5}}$$ Haskell and computation The main reason I’m writing

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