# August, 2017

## Ask Uncle Colin: Reversing Fibonacci

Dear Uncle Colin, I'm aware of Binet's formula for finding the $n$th Fibonacci number, $F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$, and wondered if there was an inverse version - to find $n$ given a Fibonacci number. -- Fibonacci Explicit Inverse, Getting Extremely Nervous But Am Understanding More Hi, FEIGENBAUM, and thank

## An Australian Dining Phenomenon

In an early draft of my forthcoming book, The Maths Behind, which will be available wherever good books are sold from September, I believe, I took the following unprovoked dig at Australia: "... it crashed into the ocean about 1,600 miles to the west of Perth, Australia. There's nothing there.

## Ask Uncle Colin: An Uncommon Logarithm

Dear Uncle Colin, I'm a bit stumped by a logs question with a variable base: $\log_{\sqrt[3]{x+3}}(x^3 + 10x^2 + 31x + 30) = 9$. I know the basics of logarithms, but this is currently beyond me. -- Obtaining Underwhelming Grade, Having To Review Every Definition Hello, OUGHTRED, and thanks for

## Wrong, But Useful: Episode 47

In this month's episode of Wrong, But Useful, we are joined by Special Guest Co-Host @jussumchick, who is Jo Sibley in real life. Colin's audio is unusually hissy in this one, which is why it's a little late; he apologises for both inconveniences. We discuss: Jo's background and work with

## Collecting coupons

For all the grief I give @reflectivemaths on Wrong, But Useful, he does occasionally ask an interesting question. In episode 45, he wondered how many packs of Lego cards one would need to acquire, on average, to complete the set of 140?1 A simpler case Suppose, instead of 140 cards,

Dear Uncle Colin, In a recent exam, I was invited to solve $12x^2 - 59x + 72=0$ without a calculator. Is that a reasonable thing to ask? Very Irate EdExcel-Taught Examinee Hi, VIETE, and I don't blame you for being cross - in a non-calculator exam, I'm not sure that

## An Integral Diversion

The brilliant @dragon_dodo sent me this puzzle: Evaluate $\int_0^1 \left(1-x^\frac{1}{7}\right)^3 - \left(1-x^\frac{1}{3}\right)^7 \d x$. I'm not going to give you the solution right now; that will come after I've rambled for a bit. After I'd solved the puzzle (see below), I wondered what each of the integrals actually evaluated to.

## Ask Uncle Colin: Spotting factors

Dear Uncle Colin, In a recent test, I stumbled across $9x^4 + \frac{1}{144x^4} + \frac{1}{2}$, which apparently factorises as $\left(3x^2 + \frac{1}{12x^2}\right)^2$. How on earth am I supposed to spot that?! - Feeling Almost Cheated, That's Only Reasonable Hi, FACTOR, and thanks for your message! I wouldn't instinctively spot that

## Revisiting some missing solutions

"You know how you're always putting things like 'just to keep @RealityMinus3 happy' in your posts?" "Of course, sensei!" "Well... you remember that post about missing solutions in a trig problem?" "Ut-oh." What follows is a guest post by Elizabeth A. Williams, who is @RealityMinus3 in real life. This thing

## Ask Uncle Colin: An Enormous Sum

Dear Uncle Colin, I've got a question that asks me to find the coefficient of $x^5$ in $(1+x)^5 + (1+x)^6 + (1+x)^7 + ... + (1+x)^{100}$. I can easily work out the coefficient in each term (it's just $\nCr{k}{5}$), but I can't see an easy way to add them up.