This is a puzzle I presented at the MathsJam conference. It’s a problem that gave me a headache for a week or so, and I thought others might enjoy it, too. I do know the answer, but I’m not going to give it away — you can tweet me @icecolbeveridge if you want to discuss your theories! (As Colin Wright says: don’t tell people the answer).
You’ve heard of the Monty Hall Problem, right?
Inspired by our Open Season post on the Perfect Cuboid earlier this year, Aperiodical reader Jos Schouten wrote to us describing his work on the problem over the past 20 years. He’s looking for someone to help take his work further. Are you up to the challenge?
Survey of the Perfect Cuboid
This article is about my search for the Perfect Cuboid (PC), which started exactly on Wednesday April 15, 1987. At that time I was a young engineer with feelings for mathematics, and employed to write C-language programs on a UNIX platform. Since then I’ve written software and explored ideas to find the cuboid, at work and at home. I still haven’t found one!
This article is also hoping to find someone in the world community as sparring partner, who likes the subject, wants to propose additional solution methods, and can help to implement such a method. The attempt will be to find a perfect cuboid with an odd side less than a googol.
Statement of the Problem
The problem is easy to state, extremely difficult to solve, but a solution, once found, would be easy to verify.
The problem in words:
Find a cuboid whose sides are integer lengths, whose face diagonals are integers and whose space diagonal (from corner to opposite corner) is integral too.
Only you can save the Wuzzit! Screenshot courtesy of Innertube Games.
Had Wuzzit Trouble been around in 2001, when I was teaching Diophantine equations… well, there wouldn’t have been an iPhone to play it on, and it would probably have been too graphically-intensive for the computers available at the time. However, I’m willing to bet fewer of my students would have fallen asleep in class.
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 $\gcd(a,b)=1$?
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?”
This is the third in a series of posts about the maths of Star Trek. Part I covered the probability of survival while wearing a red shirt, and Part II discussed the mathematics of alien biology.