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?

A friend travelling in the Eastern Caribbean sent me some photos of flowers. This lovely plant is paperflower, or lesser bougainvillea, scientific name Bougainvillea glabra.

I'm disbelieving of this being a Harvard Interview Question, but that is the title given to a question circulating on social media. It runs as follows:

It's ambiguous and that's probably why it has gone viral, with people offering a variety of answers and opinions and even the occasional quip, such as "Were they going to St Ives?".

So it's not the maths that is interesting but the impetus to debate, discuss wrangle, argue. disagree and tut about the ambiguity.

Still, to get back to the boring old maths, I thought I would ask:

If 7 is replaced by n, what is the smallest answer that we could put forward a case for, and what is the largest?

Annie L. Pressland (1862-1933), from Wikimedia

A GPS is one that "leads you down the garden path". As you read it you think you are understanding it, then it socks you on the back of the head ("This makes no sense") but finally you realise you'd been reading it the wrong way and it makes sense after all. The canonical example is

"The old man the boat."

Another one, that is really hard to parse, is

"The complex houses married and single soldiers and their families."

Eh?#%~

The Wikipedia article says the first example "has been used to test the capabilities of artificial intelligence efforts" but that was in 2016, and today DeepSeek has no trouble, it's seen it before and can analyse it with no difficulty at all

I thought I would try making up my own GPS

Here's my attempt at a GPS sentence. It took me several goes before I was able to work the idea up into something that met DeepSeek's quite stringent criteria for a bona fide example but I finally came up with this which earned its seal of approval.

"The hack saw the editor."

The sentence below (perhaps seen on a sign in a hospital) was discussed in a 1979 paper by Watson and Reich ^[1] and has attracted a good deal of interest in the intervening years.

What does it mean? Whoever wrote it presumably meant "Don't ignore a head injury however trivial it is", and that is what most people read it as, but if we unpick the wording carefully that is not what it literally means. 

Compare it with "No citizen is too poor to be taxed" which says that no citizen is so poor as to be exempt from being taxed, i.e. all citizens are taxed. So it seems the hospital sign has a meaning opposite to the intended one.

[1] https://journals.sagepub.com/doi/abs/10.1080/14640747908400750?journalCode=qjpa#:~:text=But%20No%20head%20injury%20is,too%20serious%20to%20be%20ignored.

Easy really, it’s just comma sense.

If two chords of a circle intersect  there is a neat relationship between the lengths of the four line segments created, as seen below.

This neat theorem must go back to the ancient Greek geometers. It's a fairly simple relationship but quite powerful and it often provides a short and elegant solution to geometric problems I used it yesterday to solve Think You Can Solve This? Find the Square’s Area!, posted on YouTube by "The Phantom of the Math".

The Phantom's question boils down to finding the side length of the square in the configuration below, where the square is divided by horizontal and vertical lines of symmetry as shown, the circle has its centre on the vertical symmetry line, and the circle is 3 units and 4 units away from the left-hand and top sides of the square.

The standard approach to this sort of problem is to construct a right-angles triangle, come up with expressions for the lengths of its side, then use Pythagoras to arrive at an equation which we can then go on to solve. I used this approach, then when I had an answer I went back to see how the Phantom had tackled it, and as I. expected they had followed a similar strategy.

But the problem nagged at me, because these solutions were a bit involved and mechanical and I felt there ought to be a better way. After pondering on and off for quite a while I suddenly saw a pair of intersecting chords right in front of me! Now the solution fell out really easily, like this

Finding the lengths of the four segments in terms of the side length s is not hard, we apply the intersecting chords theorem, expand and tidy up, then it's easy to solve the equation and find that s = 9 units. No applications of Pythagoras were used in this solution, and no extra lines needed to be added!

The person who invented momentum (was it Isaac Newton?) was originally going to call it 'moment' but at the very last moment they hesitated.

this winter nightfall

the trees stretch their branches up

reaching for the moon

Hail is from OE hagol, which may ultimately be related to Latin calix and Greek κάχληξ, pebble, and so from the same origin as calcium.