Then as the socks are drawn one by one from the draw it is possible that socks 1 and 2 match, socks 2 and 3 match, and so on, up to socks 2n - 1 and 2n. In all there are 2n - 1 possible pairs of consecutive socks whose colour could match.

In each consecutive pair once the colour of the first sock is known there is then a 1/(2n - 1) probability that the next sock with be the same colour, or in other words the mean expectation of the colours matching is 1/(2n - 1).

But this applies to all 2n - 1 pairs and by the principle known as linearity of expectation we can find the overall expectation simply by adding the 2n - 1 individual expectations. But (2n - 1) x 1/(2n - 1) = 1 and so we conclude the expected number of matches is 1, whatever the number of pairs of socks in the draw!

I wrote a Python program and ran it, and sure enough as predicted the result is always close to 1.

I came across in the Times "Word watch" (Times 2, 01/05/25). According to the OED it originally meant "an idler who stares at length at activity on a canal", and by extension someone who stares at anything for a long time. 

The OED classifies it as a dialect word and cites the Glossary to Bradshaw's Canals & Navigable Rivers Eng. & Wales (1904), which suggests a Lake District origin. Here is the entry, which I located with the help of Google Books

I thought I'd see if I could find out more about the origin, so I tried asking DeepSeek AI and it suggested the first print appearance as being in "The Folk-Speech of South Cheshire" by Egerton Leigh. This seemed promising! After some digging I located such a book and was getting quite excited - but alas! gongoozler was not there, DeepSeek was wrong.

The OED's summary is that the word of unknown origin, but does offer a tantalising third possibility – 

"... compare Lincolnshire dialect gawn ‘stare vacantly or curiously’, gooze (also goozen) ‘stare aimlessly, gape’."

–  and perhaps that's the best guess we have.

Here's another nice geometrical theorem. I hadn't seen it before today but it seems it's usually called Thébault's problem II and is credited to Victor Thébault (1882–1960), a mathematician who published over 1000 problems

Here's how it goes (this is a slight extension of the original problem, which had a square where the following requires only a parallelogram). Given a parallelogram ABCD as shown, erect equilateral triangles on two adjacent sides. Then the two free vertices of those triangles together with the vertex of the parallelogram opposite form an equilateral triangle.

The following is my proof. I haven't looked at any published ones, but I'm sure this is one of the standard ones and I imagine we could use complex numbers quite nicely as well, but this uses only ideas that would have been familiar to Euclid.

In this diagram there are two different lengths, marked with | and ||. Call these L₁ and L₂ and suppose the angle BCD is α. Then using the properties of equilateral triangles (all angles are 60) and parallelograms (adjacent angles add up to 180) we can mark in a series of angles.

The angle EDF is 60 + α by subtracting the other three angles at D from 360.

Now if we consider triangles EFD, FCD and BAE, we see they all have an angle of 60 + α between sides of length L₁ and L₂, which means these are all congruent and it follows immediately that the three dotted distances are equal and BEF is equilateral as claimed.

Thébault's theorem has attracted some deal of interest, because it is so neat and quite simple, but seemingly was not spotted for over 2,000 years. It has ben generalised at least twice, most recently in 2019, see The Mathematical Gazette, Vol. 103, No. 557 (July 2019), pp. 343-346.

I've just helped towards a Kitty's expensive dental bill. I was happy to do it, because I'm an Ailurophile. The word is relatively modern (1914) and comes from Ancient Greek ἀίλουρος "ailouros" = cat, which is thought to originally mean something like "wavy tail", rather poetic to my mind. A somewhat earlier term for a cat lover was Philofelist, dating from 1843.

A dog lover is a Cynophilist, or Cynophile, from Greek κυνος "kynos" = dog, from Proto-Indo-European kwon, which is also the root of words such as hound, kennel, and canine.

We found this striking Turkey Tail fungus Trametes versicolor growing on a dead log in our garden.

From Wikipedia, it is found world-wide and lives by digesting lignin, a key structural component in plants= tissues. The fungus has been used in traditional Chinese medicine and a range of beneficial properties have also been claimed for it in the setting of conventional medicine. For a summary see this article. I haven't read enough to assess how strong the evidence is, and I think opinion is probably divided.

This puzzle can be found in many places in the internet but was first published by Martin Gardner in his 'Mathematical Recreations' column in Scientific American. It goes like this (this wording is mine, not the original which I don't have to hand.)

1. Can YOU work out what the ages are? (Solution appears further down.)

2. Can you find a number different for 36 but which would still let the second mathematician deduce the ages if they knew the sum of the ages and the fact that the oldest child is called Bill?