Posted in ask uncle colin.

Dear Uncle Colin, I have a little problem. You see, there’s this bird, A, in its nest at time $t=0$ - the nest is at $(20, -17)$ - and it travels with a velocity of $-6\bi + 7\bj$ (in the appropriate units). But there’s another bird, B, whose nest is

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Posted in number theory.

My cunning plan, back last August, was sadly foiled: @christianp refused to rise to the bait. I'd written a post about finding the smallest number such that moving its final digit to the front of the number doubles its value. It turned out, to my surprise, to be 17 digits

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Posted in ask uncle colin.

Dear Uncle Colin, I’m told that the graphs of the functions $f(x) = x^3 + (a+b)x^2 + 3x -4$ and $g(x) = (x-3)^3 + 1$ touch - and I need to express $a$ in terms of $b$. Can you help? - Can’t Understand Basic Introductory Calculus Hi, CUBIC, and thanks

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Posted in dome.

I… I… I… *Looks up Ito’s Lemma* *Reaches for bargepole, then doesn’t touch it.* I… I… I… Oh! It says here, there’s a thing called Ivory’s Theorem1! What is Ivory’s Theorem? Despite the main paper I could find about it calling it “the famous Ivory’s Theorem”, it was fairly tricky

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Posted in ask uncle colin.

Dear Uncle Colin, I need to find the area between the curves $y=16x$, $y= \frac{4}{x}$ and $y=\frac{1}{4}x$, as shown. How would you go about that? Awkward Regions, Exhibit A Hi, AREA, and thanks for your message! As usual, there are several possible approaches here, but I’m going to write up

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Posted in ninja maths.

The Mathematical Ninja peered at the problem sheet: Given that $(1+ax)^n = 1 - 12x + 63x^2 + \dots$, find the values of a and n Barked: “$n=-8$ and $a=\frac{3}{2}$.” The student sighed. “I get no marks if I just write down the answer.” Snarled: “You get no

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Posted in ask uncle colin, big in finland.

Dear Uncle Colin, A seven-digit integer has 870,720 as its last six digits. It is the product of six consecutive even integers. What is the missing first digit? Please Reveal Our Digit! Underlying Calculation Too Hi, PRODUCT, and thanks for your message! There are several approaches to this (as usual)

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Posted in geometry.

Once upon a MathsJam, Barney Maunder-Taylor showed up with a curious object, a wedge with a circular base. Why? Well, if you held a light above it, it cast a circular shadow. From one side, the shadow was an equilateral triangle; along the third axis, a rectangle. A lovely thing.

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Posted in ask uncle colin.

Dear Uncle Colin, I have to show that $\Pi_1^\infty \frac{(2n+1)^2 - 1}{(2n+1)^2} > \frac{3}{4}$. How would you do that? Partial Results Obtained Don’t Undeniably Create Truth Hi, PRODUCT, and thanks for your message! That’s a messy one. I can see two reasonable approaches: one is to take the whole thing

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Posted in geometry.

On a recent MathsJam Shout, an Old Chestnut appeared (in this form, due to @jamestanton): If you’ve not seen it, stop reading here and have a play with it - it’s a classic puzzle for a reason. Below the line are spoilers. Counting is hard The first thing you’d probably

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