grunt -> hunt, grow -> how, ground -> hound, grandstand -> handstand, etc.

I noticed that a remarkable number of words starting with gr are still words if you swap the gr for h. For example, the words in the title of this post. How many words is this true for? Which pair of prefixes has the most words in common?

Here’s a Python script I wrote to answer those questions. Here’s the list of words I used. And here are the results. I only looked at prefixes of one or two letters.

The best pair was no and u. Here’s the list of suffixes they have in common. Most of it is words which can be prefixed with un or non. That isn’t very interesting, so I think the real winner is (b,st), with 1085 suffixes in common. It’s the first pair where one of the prefixes is two letters, and where most of the words aren’t just words with another Latin prefix in front of them.

I could do loads of calculations like this. If you import the Python script as a module, you can have a look at all the data it computes. Very interesting!

The word list I used probably skewed the results quite a bit because it contains lots of words which are conjugations or pluralisations or whatever of the same root word, as well as a load of really weird words which probably occur once in the whole corpus. I think if I look at this again I’ll use something like this frequency list, and use the frequencies of words as a weighting for scoring prefix-pairs.

An infinite series problem

A couple of undergrads came into maths-aid today with a problem: does the series $\sum_{n=1}^{\infty} \frac{1}{n \ln(n+1)}$ converge?

Hardy’s Course of Pure Mathematics

A while ago I bought a copy of Hardy’s “A Course of Pure Mathematics” for a fiver. I think I must’ve got it from Westwood Books ((Westwood Books has a really good collection of maths books – one of the owners is an ex-mathematician, and he finds his way to be in just the right place when departments clear out their libraries. I found a book by the only other mathematising Perfect I know of at Westwood. Great place!)) in Sedbergh.

Anyway, my imagination has completely run dry for the moment, so I’ve decided to do some exercises from the book to pass some time. Maybe this could become a series of posts?

Hardy says in the preface,

This book has been designed primarily for the use of first year students at the Universities whose abilities reach or approach something like what is usually described as `scholarship standard’.

I regard the book as being really elementary.

Nothing taxing, then. It occurs that I’m opening myself up to accusations of sloppy thinking and gross inaptitude at mathematics, but I suppose I’m better off knowing if that’s the case.