A nice puzzle from reddit: Evaluate $\sqrt{\sqrt{2025}-\sqrt{2024}}$ The suggestion is that this is the sort of thing that’ll be in Olympiad papers in a couple of years. Fair enough. This isn’t quite a real-time solution, but it’s roughly how I thought about it and tackled it. Spoilers are (as you might surmise) below the line. The first thing I notice is that 2025 is a square number (it’s $45^2$). That means 2024 is $(44)(46)$, although, as it turns out, that’s not immediately helpful to me. Instead, I followed a problem-solving strategy that was crystallised to me by @colinthemathmo: “make ..read more

Today’s problem: $\frac{a}{b} = \frac{2}{3}$ $a^b = b^a$ Find $b-a$. I’m just going to straight-up answer this below the line. My first step would be to separate the $a$ and $b$ in the second equation, to get $a^{1/a} = b^{1/b}$. I was tempted by logs for a moment, but we don’t know that $a$ and $b$ are positive. I can also separate the first equation to get $b = \frac{3a}{2}$. Now I have $a^{1/a} = \left(\frac{3a}{2}\right)^{\frac{2}{3a}}$. Now we’re cooking. Raise both sides to the power of $3a$, which is definitely the ugly thing: $a^3 = \left(\frac{3a}{2}\right)^2$ – and we’re almost the ..read more

You have 20 spaces, which you want to fill with numbers, in order. You will be given 20 random numbers between 0 and 999 (inclusive), each of which you must place into a space as soon as you see it. If you can place all 20 while keeping them in order, you win. What’s your best strategy? How likely are you to win? I’ve done some analysis, which I’ve put below the line because it might constitute a spoiler. For me, the trick to analysing this game is to realise that every move splits a group of spaces into two parts, one or both of which may be empty. The probability of being able to successfull ..read more

Dear Uncle Colin How would you prove that the area under the curve $y=\sin(x)$ from $x = 0$ to $x=\pi$ was exactly 2? ‘Cause I Realised Calculus Lacked Explanation Hi, CIRCLE, and thanks for your message! The standard way is just to know that the integral((with respect to $x$)) of $\sin(x)$ is $-\cos(x)$, plug in the limits and dust your hands as if you’ve done something clever. But – as you suggest – that’s more of a trick than an explanation. Let’s look instead at a unit circle, and consider a tiny sector of it. The angle at the centre is $\Delta\theta$, and the whole thing is raised at an ..read more

Via reddit, a challenge to solve: $y = y’ + y’’ + y’’’ + \dots$ Once you’ve stopped running away and hiding, I’ll show you the solution they suggested. The trick is to notice that $y’ = y’’ + y’’’ + y^{(4)} + \dots$, so $y = y’ + (y’)$, which is straightforward to solve: $y = 2y’$ $y = Ae^{x/2}$. It’s a good idea to check that it works: $y’ = \frac{1}{2}y$ $y’’ = \frac{1}{4}y$ $\dots$ The right-hand side is then $y\left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8}+\dots\right)$, and the bracket evaluates to 1. Boom. But wait a moment We’ve found a solution. How do we know it’s the only one? Wha ..read more

In the course of solving a puzzle, I had cause to assert that $\pi <\sqrt{10}$. I mean, that’s just true: I know that $\sqrt{10} \approx 3.16$ and $\pi \approx 3.14$; I also know that $\pi < \frac{22}{7}$ and that $\left(\frac{22}{7}\right)^2 < 10$. But these beg the question. How would I go about proving the inequality without resorting to “I just know”? Polygons My first thought is, can I construct a polygon with a perimeter of $\sqrt{10}$ that lies outside of the unit circle? An $n$-gon is made up of isosceles triangles, each of which has an angle at the apex of $\frac{2\pi}{n}$ an ..read more

A post on Hacker News descended into an argument in the comments ((finish the sentence before you go “oh, quelle surprise”, please.)) about whether random(random()) and random() * random() gave different distributions. Now, I know better than to wade into a Hacker News discussion unless it’s about mental arithmetic and can name-drop Colin Wright. However, this did interest me enough to think about. Maybe you’d like to think about it, too. Spoilers below the line. At heart, the question asks “is drawing a random number $X\sim U[0,1]$ and then another $Y\sim U[0,X]$ the same thing as drawing two ..read more

A trick I learned from Colin Wright: Take the first nine powers of 2 and place them in lexical order; then put a decimal point after the first digit. OK, Colin, here you go: 1.28 1.6 2. 2.56 3.2 4. 5.12 6.4 8. These numbers are approximately $10^{0.1}$, $10^{0.2}$, etc., up to $10^{0.9}$. Huh! Isn’t that nice? Why does that work? You might want to have a think about it. I’ll spoiler it below the line. Well, it boils down to the handy numerical coincidence that $2^{10} \approx 10^3$. That means that $2 \approx 10^{0.3}$, $4 \approx 10^{0.6}$ and $8 \approx 10^{0.9}$, before we get into an ..read more

Dear Uncle Colin, I have to work out 17.5% of 354 in my head. How would you go about it? Various Arithmetical Tricks Hi, VAT, and thanks for your message! The traditional method is to take 10% of your base number (35.4), then add on half (17.7) and half again (8.85) to get 61.95. However, that’s quite a lot to keep in your head at once. Alternatively, you can shuttle factors: 17.5% of 354 is 35% of 177, or 70% of 88.5. That’s still fairly tricky, but not too hard to reach 61.95. My favourite method is to note that 17.5% is $\frac{7}{40}$, which can be written as $\frac{1}{6} + \frac{1}{120 ..read more

On reddit, somebody asked: why does $\frac{7}{3} + \frac{7}{4} = \frac{7}{3}\times\frac{7}{4}$? My first thought: that’s a neat variation of $2+2=2\times 2$. My second: when does that hold true? It’s not too tricky to solve algebraically: $x + y = xy$ $x = xy - y$ $x = y(x-1)$ $y =\frac{x}{x-1}$ $y = 1 + \frac{1}{x-1}$. That’s clearly true for $x=y=2$, and less clearly true (but still true) for $x= \frac{7}{4}$. So that’s neat: we can construct pairs of numbers that work for any $x$ except for 1. I also wondered whether it turned out more neatly if, instead of starting from 0, we started fro ..read more