### Cushing your luck: properties of randomly chosen numbers

Long-time Aperiodical muse David Cushing has made a bet with us that he can give us an interesting post every Friday for the next ten weeks. Every week that he sends a post, we buy him a bar of chocolate. Every week that he doesn’t send us a post, he buys us a bar of chocolate. For his first trick, David is going to do some unnatural things with the natural numbers.

The greatest common divisor (gcd) of two or more integers is the greatest integer that evenly divides those integers. For example, the gcd of $8$ and $12$ is $4$ (usually written as $\gcd(8,12)=4$). Two integers are called coprime (or “relatively prime”) if their gcd is equal to $1$.

A reasonable question to ask is,

Given two randomly chosen integers $a$ and $b$, what is the probability that $\gcd(a,b)=1$?

### Axes to Axes

In which the intrepid maths-crime-fighting duo of Gale and Beveridge find themselves thrust back to a time before people could do maths properly.

It had been a quiet night at the Aperiodical police station. Apart from a few cases of broken scheduling in Excel formulas – nothing a bit of TIME() in the cells wouldn’t put right – there was nothing.

At 11pm, the phone rang. I looked at Sergeant Gale. Sergeant Gale pointedly looked at the phone, raised an eyebrow, and returned to his sudoku.

“Maths Police, bad graphs department. Constable Beveridge speaking, how can I help?”

### The Maths of Star Trek: The Original Series (Part III)

This is the third in a series of posts about the maths of Star TrekPart I covered the probability of survival while wearing a red shirt, and Part II discussed the mathematics of alien biology.

### The Maths of Star Trek: The Original Series (Part II)

This is the second in a series of posts about the maths of Star Trek. Part I covered the probability of survival while wearing a red shirt.

### The Maths of Star Trek: The Original Series (Part I)

As you may well know, Star Trek was a science fiction TV show in the late 1960s. It featured futuristic technology and science fiction ideas such as warp drives, transporters, strange new worlds, time travel, and green alien space babes. And the possibility of all these things has, in the past, been discussed by experts, and nerds, in great detail. Especially that last one about green space babes.

But dammit, I’m a mathematician, not a physicist. So, instead of talking about the science of Star Trek yet again, what about the maths of Star Trek? After all, Star Trek is science fiction, but there is no such thing as maths fiction – so any mathematics featured on the show is sure to be on firmer ground. Right? Or as Spock himself says in ‘The Conscience of the King’;

SPOCK: Even in this corner of the galaxy, Captain, two plus two equals four.

Should we even expect much maths to feature on a simple space adventure show? In fact, many interesting mathematical ideas were raised during the show’s short run of 79 episodes, including; the probability we are alone in universe; a paradox that upset 20th century mathematicians as well as 23rd century androids; the mathematics of alien and Earth biology; and the most important question of all – when on a dangerous away mission, does the colour of your shirt really affect your chances of survival?

### The perfect formula for mathsiness

It’s mid-January, which means it’s time for the tabloids to trot out their annual “this is the most miserable day of the whole year!” story — before they spend the rest of the year blaming immigration, youth and political correctness for problems they’ve spent the last year stoking up.

Ahem.

### AS maths results and batting averages

Phil Harvey gave a talk on this subject at last November’s MathsJam conference. We liked it so much, we asked him if we could put it on the site. Phil’s kindly written his talk up as an essay for us.

I am 64¼ years old and I’ve been a maths teacher all my working life. In that time things have changed. Long gone are the days when gowned masters would sweep in, silence any murmur with half a raised eyebrow, and delight compliant uniformed schoolchildren with chalk-covered boards of mathematical exposition.

No, you’re right. That never happened outside the covers of Goodbye Mr Chips, even in my day.

The reality then. Schoolchildren have morphed into learners. Exam results rule. Quality (in the sense that Orwell might have used the word) is managed by quality managers. And so our working lives are driven by the pursuit of Ofsted targets, success rates, achievement rates, benchmarks, observation grades, results. And every joyless lesson has its own lesson plan, with aims, objectives, learning outcomes and action points. But above all, those damned results – and every year, year after year, they had to IMPROVE.

Well I was no good at any of this stuff – and consequently I always got on very badly with my managers. Until one year…