Posted in logarithms, ninja maths

“Here’s a quick one,” suggested a fellow tutor. “Prove that $2^{50} < 3^{33}$.” Easy, I thought: but I knew better than to say it aloud. First approach “I know that $9 > 8$,” I said, checking on my fingers. “So if $2^3 < 3^2$, then $2^{150} < 3^{100}$ and $2^{50}

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Posted in ask uncle colin

Dear Uncle Colin, How can I tell whether $\frac{221}{391}$ and $\frac{403}{713}$ are equivalent? - Calculator Answer Not Considered Enough, LOL Hi, CANCEL, and thanks for your message! There’s a naive way to do it and a clever way. Let’s do it naively The naive way is to see whether $\frac{221}{391}

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Posted in probability, puzzles

In a recent MathsJam Shout, courtesy of Bristol MathsJam, we were given a situation, which I paraphrase: Cards bearing the letters A to E are shuffled and placed face-down on the table. You predict which of the cards bears which letter (You make all of your guesses before anything is

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Posted in ask uncle colin

Dear Uncle Colin, I’m trying to solve $2\cos(3x) = -\sqrt{2}$, for $0 \leq x \lt 2\pi$, but the answers I find are outside the specified interval, and obviously I miss the ones that are in the interval. How would you tackle this? - Like A Puzzle, Like A Cosine Equation

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Posted in dome

What it is Every so often, one comes across a teacher who is Properly Evil. I’ll spare names here, but I have a clear, strong memory of being introduced to the Collatz conjecture on a school trip. “Take a number, let’s say 3. If it’s odd, you treble it and

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Posted in ask uncle colin

Dear Uncle Colin, I’m trying to organise a tournament involving seven teams and two pitches. The following conditions must hold: Each team plays four games No pair of teams meets more than once Each team must play at most one pair of back-to-back matches How would you solve this? Bit

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Posted in Uncategorized

This is based on a puzzle I heard from @colinthemathmo, who wrote it up here; he heard it from @DavidB52s, and there the trail goes cold. The Mathematical Ninja lay awake, toes itching. This generally meant that a mission was in the offing. Awake or dreaming? Unclear. But the thought

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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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