# Browsing category ask uncle colin

Dear Uncle Colin, In a culture where people have children until they have a daughter and then stop, what proportion will be daughters? * Families Retain Extra Girls Evidently Hi, FREGE, and thanks for your message1 ! Spoiler alert: it’s a half. There are several ways to see this, and

## Ask Uncle Colin: The Polar Express(ion)

Dear Uncle Colin, I have the polar equation $\theta = k$, and I want to turn it into rectangular coordinates. I worked it out to be the line $y = x \tan(k)$, but the solution says it’s only a half-line. Why is that? Perplexed Over Line And Radius Hi, POLAR,

## Ask Uncle Colin: A Six-Digit Square

Dear Uncle Colin, I’m told a six-digit number, ABC,DEF1, is a perfect square such that DEF is 8 times ABC. How do I find this number? - Very Interesting, Except That Eight Hello, VIETE, and thanks for your message! This is as much of a typographical challenge as it is

## Ask Uncle Colin: A round-robin

Dear Uncle Colin I’m organising a tournament with 16 teams, and wanted to arrange it in five stages, each consisting of four groups of four teams. However, I found that after three rounds, it wasn’t possible to find any groups without making teams play each other again! Why is that?

Dear Uncle Colin, I have to solve $5^x = 6 - 5^{1-x}$ - I understand it’s going to end up as a quadratic, but I can’t see how! - Explain, Uncle Colin, Like I Demand! Hi, EUCLID and thanks for your message! The key thing here is to spot that

## Ask Uncle Colin: Traffic Flow

Dear Uncle Colin, I read that when cars are driving at 70mph on the motorway, they take up more space than when they travel more slowly (because you need to leave a longer safe gap between them). What’s the most efficient speed for motorway travel if you want to get

## Ask Uncle Colin: A Fishy Derivative

Dear Uncle Colin, I have the equation of a curve, $\frac{2x+3y}{x^2 + y^2} = 9$. If I differentiate implicitly using the quotient rule, I get $\diff{y}{x} = \frac{2(x^2 + 3xy - y^2)}{3x^2 - 4xy - 3y^2}$. If I rearrange first to make it $2x + 3y = 9\left(x^2 + y^2\right)$,

## Ask Uncle Colin: Scheduling a tournament

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