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

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Posted in algebra.

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

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Posted in algebra.

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

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Posted in algebra.

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

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Posted in algebra, ask uncle colin.

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

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Posted in algebra, big in finland, probability, puzzles.

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

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Posted in algebra, ask uncle colin.

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

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Posted in algebra.

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...

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Posted in algebra, ask uncle colin, fractions.

Dear Uncle Colin, I recently had to decompose $\frac{3+4p}{9p^2 - 16}$ into partial fractions, and ended up with $\frac{\frac{25}{8}}{p-\frac{4}{3}} + \frac{\frac{7}{8}}{p-\frac{4}{3}}$. Apparently, that's wrong, but I don't see why! -- Drat! Everything Came Out Messy. Perhaps Other Solution Essential. Hi, there, DECOMPOSE, and thanks for your message - and your

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Posted in algebra.

Usually, when faced with a word problem, I take the most obvious approach and call it done. But then, sometimes, I read of an alternative approach that makes me go "Whoa." This is one of those times. Here's the problem: One day, a person went to a horse racing area.

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