# Report on fortnightly maths prize question for 26 September

The fortnightly maths prize question for 26 September was:

5=22+12 and 13=32+22

Make a good guess for a rule about which odd prime numbers can be broken down this way, into a sum of two squares. Make a good guess about which whole numbers generally (not just primes) can be broken down into a sum of two squares. Continue reading

# A problem with Edexcel about Quicksort

Edexcel are insisting that the order of items within each new sublist remains the same as in the previous list, though without giving any mathematical reason why this is even a good idea, let alone essential. Continue reading

# Why do we choose the middle item as the pivot in Quicksort?

Quicksort works best when the pivot divides the items into equal groups. Choosing the middle item from the unsorted list does not guarantee that. But there is a reason why a rule of choosing the middle item as pivot is a workable, simple, not-too-bad rule. Continue reading

# What if we have items equal to the pivot, neither bigger nor smaller?

It’s unlikely you’ll have something like that in the exam. But if you do, just do what the textbook says: consistently read “smaller than or equal to the pivot, but not actually the pivot item itself” for “smaller”, or “bigger than or equal to the pivot, but not actually the pivot item itself” for “bigger”. One or the other. Doesn’t matter which.

# How Quicksort can be done “in place”

Quicksort can be done “in place”, just by swapping items within the same list, without creating another list-space into which items from the previous list are moved.

# How proof dropped out of school maths

It is a very odd choice by Edexcel to have FP1 (Further Pure Mathematics 1) students study proof by induction, a particular (and rather subtle and paradoxical) form of proof, before they’re introduced to proof more generally. And it raises the question – just when and how proof disappeared from school maths. As far as I can make out, the story goes something like this. Continue reading