# Browsing category ask uncle colin

## Ask Uncle Colin: A Missing Digit

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)

## Ask Uncle Colin: An Infinite Product

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

## Ask Uncle Colin: Why is $e$ not 1?

Dear Uncle Colin, If $e = \left( 1+ \frac{1}{n} \right)^n$ when $n = \infty$, how come it isn’t 1? Surely $1 + \frac{1}{\infty}$ is just 1? - I’m Not Finding It Natural, It’s Terribly Yucky Hi, INFINITY, and thanks for your message. You have fallen into one of maths’s classic

## Ask Uncle Colin: Curved Surface Areas

Dear Uncle Colin, I’ve been struggling with this: “If the surface area of a sphere to cylinder is in the ratio 4:3 and the sphere has a radius of 3a, calculate the radius of the cylinder if the radius if the cylinder is equal to its height.” Can you help?

## Ask Uncle Colin: A problem of squares and powers

Dear Uncle Colin, I have to solve $615 + x^2 = 2^y$ for integers $x$ and $y$. I’ve solved it by inspection using Desmos ($x=59$ and $y=12$ is the only solution), but I’d prefer a more analytical solution! Getting Exponent Right Makes An Interesting Noise Hi, GERMAIN, and thanks for

## Ask Uncle Colin: A Binary Fraction

Dear Uncle Colin, How would you write $\frac{1}{10}$ in binary? Binary Is Totally Stupid Hi, BITS, and thanks for your message! I have two ways to deal with this: the standard, long-division sort of method, and a much nicer geometric series approach. Long division-esque While I can do the long

## Ask Uncle Colin: Round-Robin Progress

Dear Uncle Colin, I lost the first game of my Big Internet Math-Off tournament - can I still win the group and qualify for the semi-finals? - Surely Combinations Of Talent, Luck And Nous Deliver? Hi, SCOTLAND, and thanks for your message! Because the tie-break rules aren’t currently clear, I

## Ask Uncle Colin: Some Symmetric Algebra

Dear Uncle Colin, If I know that $a+b+c = 0$, how can I show that $(2a-b)^3 + (2b-c)^3 + (2c-a)^3 = 3(2a-b)(2b-c)(2c-a)$? - Something You Might Merrily Explain? Thanks! Regards! Yippee! Hi, SYMMETRY, and thanks for your message! As usual, there are myriad ways to attack this, of which I

## Ask Uncle Colin: A Cosine Proof

Dear Uncle Colin How would you prove that $\cos(2x) > 1 - 2x^2$ for $x > 0$? - Can’t Obviously See It, Need Explanation Hi, COSINE, and thanks for your message! I’d start by noting that as soon as $x > 1$, the right hand side is smaller than -1,

## Ask Uncle Colin: A Modest Inheritance

Dear Uncle Colin, My six siblings and I have inherited a fortune of £$10^{10} + 10^{(10^2)} + \dots 10^{(10^{10})}$, to be divided evenly between us. However, we’re a very squabbly family, so we want to know how much money will be left over once it’s divided up. Can you help?