Welcome to the sixth edition of the Carnival of Mathematics! I’ve certainly enjoyed looking through this fortnight’s submissions, and hope you will too.

Trivium Pursuit‘s series on when best to introduce formal arithmetic to children continues with this article, setting out their arguments for providing an informal experience of mathematics before the age of ten. I remember plodding through worksheets at that age, where “harder” simply meant “more tedious”, perfoming larger calculations with by-then familiar processes; it’d be interesting to know how my mathematical career would have been shaped by avoiding such drill in favour of the suggested approach. I suspect most mathematicians will also find it a very strange idea, assuming they also took to formal arithmetic at a very young age and thus find it hard to imagine struggling with such material.

Of course, the beauty of mathematics often lies not in getting an answer, but getting there in a clever way. Political Calculations presents criss-cross multiplication; an easier way to multiply two digit numbers in your head; whilst Let’s play math offers some guidelines for K-12 students to help organise their thoughts in problem solving. But over in the universe of discourse, Mark Dominus warns of the danger of over-thinking a problem where brute-force may suffice. This is particularly true when computers are thrown into the mix, as the time taken to devise and program an elegant solution may well exceed the runtime of the naive approach.

At the other extreme, though, there are computations which you wouldn’t survive to see the result of if you took the brute-force option. I’m reminded of this e2 writeup, which describes a particularly daft hello world program along with a curious question: if hardware performance continues to double with Moore’s law, how long should you wait before running such a program? After all, if you estimate a function will take 4 years to execute on your current machine, but one twice as fast will be available in 18 months, then you can save 6 months (and a load of electricity) by waiting a year and a half, obtaining the faster kit, and running your program for just two years. Conversely, if you want to guard against brute-force attacks for a given length of time, just how good does your encryption have to be?

Also on e2 (hey, if I’m hosting, I can plug my favourite site, right?), sam512 has been exploring very, very large numbers, inspired by this edition of xkcd: so far the clarkkkkson, hyper operator and linear array notation have been covered.

Many of this weeks submissions presented problems, of varying difficulty. As a warm-up, Sharp Brains‘ new puzzle master offers a brainteaser entitled ‘Party For Polyglots’. More demanding are two entries from MathNotations, aimed at calculus students: the first concerning properties of the ellipse, the second, more advanced post is on exploring infinite series. We close with the most amibitious material, the Unapolagetic Mathematician‘s examination of the knot colouring problem: for the background, see here; this week’s submission explains what’s going on. I feel that I had an unfair advantage since my flatmate is a knot-theorist; I’m often intrigued by the techniques applied to this field. Plus I rarely get to draw pictures during my own research!

Thanks to all who submitted content, and to those of you reading this! The next stop for the carnival is Nonoscience, on the 4th of May.