# Browsing category algebra

## The Involution of Polynomials

Last time out, I looked at a problem unearthed by @mathsjem - to find the cube root of a degree-six polynomial. This led (unsurprisingly) to a quadratic: $3 + 4x - 2x^2$. When checking whether this was indeed the answer, I hit a problem: is there a simple way to

## The Evolution of Polynomials

It's always fascinating to see what's going on in textbooks of the olden days, and National Treasure @mathsjem recently found a beauty of its type. Look at those whences! Check out the subjunctives! It thrills the heart, doesn't it?1 What caught my attention, though, was evolution - in this context,

## A RITANGLE problem

When RITANGLE advises you to use technology to answer a question, you know it's going to get messy. So, with some trepidation, here goes: (As usual, everything below the line may contain spoilers.) It's easy enough to do this in Geogebra - but somehow a little bit unsatisfactory to move

A charming little puzzle from Brilliant: $x^2 + xy = 20$ $y^2 + xy = 30$ Find $xy$. I like this in part because there are many ways to solve it, and none of them the 'standard' way for dealing with simultaneous equations. You might look at it and say

## Lines and squares

This puzzle presumably came to me by way of @ajk44, some time ago. Thanks, Alison! The problem, given here, is to find the equations of two lines that complete a square, given: Two of the lines are $y=ax+b$ and $y=ax+c$ One of the vertices is at $(0,b)$. The example given

## Mishandling polynomials for fun and profit

One of the more surprising results a mathematician comes across in a university course is that the infinite sum $S = 1 + \frac{1}{4} + \frac{1}{9} + ... + \frac{1}{n^2} + ...$ comes out as $\frac{\pi^2}{6}$. If $\pi^2$s are going to crop up in sums like that, they should be

## 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

## A coin sequence conundrum

Zeke and Monty play a game. They repeatedly toss a coin until either the sequence tail-tail-head (TTH) or the sequence tail-head-head (THH) appears. If TTH shows up first, Zeke wins; if THH shows up first, Monty wins. What is the probability that Zeke wins? My first reaction to this question

## Ask Uncle Colin: A STEP in the right direction

Dear Uncle Colin, I'm struggling with a STEP question. Any ideas? Given: 1. $q^2 - pr = -3k$ 2. $r^2 - qp = -k$ 3. $p^2 - rq = k$ Find p, q and r in terms of k. - Simultaneous Triple Equation Problem Hi, STEP, and thanks for your

## Mr Penberthy’s Problem

It turns out I was wrong: there is something worse than spurious pseudocontext. It's pseudocontext so creepy it made me throw up a little bit: This is from 1779: a time when puzzles were written in poetry, solutions were assumed to be integers and answers could be a bit creepy...