# Browsing category ninja maths

## Why $\phi^n$ is nearly an integer

This article is one of those 'half-finished thoughts' put together late at night. Details are missing, and -- in a spirit of collaboration -- I'd be glad if you wanted to fill them in for me. The estimable @onthisdayinmath (Pat in real life) recently posted about nearly-integers, and remarked that

## Brutal simultaneous equations

I recently became aware of the IYGB papers, available from Madas Maths. Like the Solomon papers, they're intended to stretch you a bit -- they're ranked by difficulty from standard to extremely hard. My student, being my student, demanded we go through one of the extremely hard ones. There were

## On the square root of a third

While I'm no Mathematical Ninja, it does amuse me to come up with mental approximations to numbers, largely to convince my students I know what I'm doing. One number I've not looked at much1 is $\sqrt{\frac{1}{3}}$, which comes up fairly frequently, as it's $\tan\left(\frac{\pi}{6}\right)$2 . Ninja-chops taught me all about

## The Mathematical Ninja and the twenty-sixths

The Mathematical Ninja played an implausible trick shot, not only removing himself from a cleverly-plotted snooker, but potting a red his student had presumed safe and setting himself up on the black. Again. "One!" he said, brightly, and put some chalk on the end of his cue. The student sighed.

## The Mathematical Ninja and The Slinky Coincidence

"No, no, wait!" said the student. "Look!" "8.000 000 072 9," said the Mathematical Ninja. "Isn't that $\frac{987,654,321}{123,456,789}$? What do you think this is, some sort of a game?" "It has all the hallmarks of..." "I'll hallmark you in a minute!" said the Mathematical Ninja. Seconds later, the students arms

## Ask Uncle Colin: rearranging $\cos^3(x)$

Dear Uncle Colin, I recently came across a problem in which I had to integrate $\cos^3(x)$. Somewhere in my mind, I recall that the thing to do is to make it into something involving $\cos(3x)$, but I couldn't put the details together. Could you help? -- Not A Very Inspired

## The Mathematical Ninja finds the value of $\pi$

This article has also been published as part of the Relatively Prime zine. The student yelped, and found his wrists and ankles strapped to the wheel before the lesson had even started. "Good morning," said the Mathematical Ninja. "I think you and I need to have a little... talk." The

## The Mathematical Ninja and the First Rule of $\phi$ club

The Mathematical Ninja placed his quadruple espresso on the table; @dragon_dodo looked up from her laptop and smiled. "You're taking it easy on the caffeine this morning, I see?" The Mathematical Ninja nodded. "Yeah, this is only my third cup. What are you working on?" "Computing assignment." The Mathematical Ninja,

## How the Mathematical Ninja approximates factorials, revisited

"That @ColinTheMathmo chap had a blog post on Stirling's approximation, too," said the student, spotting a chance to move the lesson away from his disappointing mock exam results. "Used it to work out 52!" "I saw it," said the Mathematical Ninja, polishing his weaponry smugly. "It... wasn't bad, exactly..." "But

## How the Mathematical Ninja estimates factorials

"I suppose," said the Mathematical Ninja, "I can allow you to put $20!$ into a calculator. There's absolutely no reason you should know that it turns out to be about $2.4 \times 10^{18}$." The student tapped the numbers in, frowned, thought for a moment and said "OK, I'll bite. How...?"