# MATH PROBLEMS?

Maths in the City posted this on twitter:

In order to make a number we can call, we need both of $n=\left(10x\right)\left(13{i}^{2}\right)$ and $m=\frac{\mathrm{sin}\left(xy\right)}{2.362x}$ to be integers.

# Putting all the world’s water in buckets

Following this pair of tweets about water:

The obvious question is, at what point are the two numbers the same? Or,

If you put all the Earth’s water into containers of the same size so that each container carries as many atoms of water as there are containers, how big is each container?

# The sign on my office door

Recently, someone left my office at Newcastle University and a new person took their place, so we needed a new sign on our front door. I wanted to do something clever with it, but it needed to be instantly legible to lost supervisors trying to find their students.

My first thought was that since there are seven of us, something to do with the Fano plane would look good. Our names didn’t have enough of the right letters in the right places for it to work, though.

That got me thinking about the Levenshtein distance. The Levenshtein distance between two strings is a measure of how many changes you need to make to one to end up with the other. I wrote a Python script which calculated the distance between each pair of names:

# Personal opinions on LaTeX

I’m editing a paper (12 months and counting!) and I’ve had a few thoughts about LaTeX that I thought I’d write down. I don’t even care if this makes me a neckbeard, that’s the mood I’m in currently.

The hyperref package makes your references clickable when you compile to PDF. I can’t think of a reason not to use it.

I dislike people who set their editors to have a fixed maximum line-width! Word-wrap works fine, and means the window is full no matter how big it is. Also, newlines can be used to separate thoughts more clearly.

# 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.

# A gimmicky Normal PDF visual aid

Quite a lot of people have trouble visualising what’s going on when you look up p-values in a table of the normal CDF. I had an idea about how to explain a bit more dynamically using a bit of Craft Time inegnuity.

I made this sketch of the normal PDF in Asymptote and printed out two copies. I cut out the area under the curve of one of them, and glued its corners to those of the other one.

Now I can slide sheets of coloured paper between the two copies of the graph, showing visually the area under the curve up to a certain limit. Using a second colour allows me to show how the area between two values is calculated by subtracting the area corresponding to the lesser value from the area corresponding to the greater value.

# A mathematical toy made in Monkey

This is a thing I made in Monkey.

It’s quite nice.

If you’re using Internet Explorer 8 or older, it won’t work. Sorry.

# Some awful maths jokes

I’ve been annoying Nathan with these all week.

Why is ${D}_{6}$ better for the environment than ${Z}_{6}?$ Because it doesn’t commute.

Which bit of Jessop’s does René Descartes like best? The Konica section.

What goes well with a curry and can compute any recursive function? NAND bread.

A colleague claims to know the secret recipe for authentic Coca-Cola. How does he proves this to you without revealing the recipe? With a zero-knowledge protocola.

In the Music Hall of Fame, which guided visit takes you from James Brown up to George Clinton in the quickest time, but is best enjoyed while under the influence? The add-joint funk-tour.

# Disappointing Stationery

So I was just making my dad’s birthday card. I felt it would benefit from some typography, so I got out a sheet of letter transfers I bought a while ago.

Something immediately leapt out at me: where are all the Es? As anyone who was a bit obsessed with cryptography when they were young knows, the most common letter in English is E, by a long way, and there are hardly any Qs, Xs or Zs. This fact has helped us (and our enemies) win wars.

So, here’s a graph of the relative letter frequencies on the sheet compared to the frequencies in the English corpus (from Simon Singh’s site):

There are about half as many Es as there should be and loads more Qs, Xs and Zs than I will ever use. Now, the sheet’s quite small, so a statistician would do a hypothesis test to see if there is a significant difference between the sheet and the corpus, but I’m not one of those so I’m going to stop here, outraged at WH Smith’s lack of rigour.

I suppose if they’d only put one Z in people would worry about a situation in which they needed to use two Zs, while it isn’t as immediately apparent that there aren’t enough Es to make use of the whole sheet.

The upshot is, I think my dad’s card is going to have to have somebody sleeping in a queue for a xylophone quorum, in order to get the most out of this sheet. My hands are tied by Statistics.

PS It isn’t all bad though – there are twice as many 1s as any of the other non-zero digits, so it at least makes a nod to Benford’s Law.

# Interesting papers collection

I've started a collection of interesting non-pregroup-related papers on Mendeley. At the moment this includes a formal system which makes the diagrams in Euclid's proofs a part of the formal reasoning, and "How to explain zero-knowledge protocols to your children"