30 September, 2025

Anglesey has what may be the finest display of cyclamens in the country, rivalled only by the RHS gardens at Wisley.

The name cyclamen goes back to Greek kyklos, "circle", and we have related words like circus and cycle etc. These words may even be related to ring, the common PIE root being something like *sker-, "turn", with kyklos being a reduplicated form skersker. "ring" is from the same PIE root, so I suppose "circus ring" could be consider a triplication.

But what has this to do with cyclamens? Well the plant grows from a round tuber and the plant is named for that, the theory goes. This seems a bit tenuous to me; many plants have round tubers or bulb or corms, so why are cyclamens special? 

I saw this on "Andymath" but I haven't looked at the solution there. The problem is to find the marked angle.

We have two equilateral triangles, not necessarily of equal size, arranged as shown in the diagram, but no information about the orientation of the upper triangle or its relative size.

One reason geometry problems are so interesting is that often the solution requires some divergent thinking, "out of the box" as the saying goes. This can often involve adding auxiliary elements such as extra lines or circles to the given diagram, and seeing what to add can require considerable creativity.

But experience plays a role too, and one rule of thumb is that it is usually worth drawing in any standout perpendiculars and seeing if they do anything for us. So let's add line CD.

Now we notice that angles CGD and CBD are both interior angles of equilateral triangles, so they are equal. They both stand on CD and the converse of the angles in the same segment theorem tells us CD must be a chord of a circle passing through BG and B. Let's add that circle.

But now we see DB is a chord in that circle with angles DCB and DGB being angles in the same segment and therefore equal. DCB is obviously 30 degrees and so the angle DGB that we want is 30 degrees also. Solved!

A different approach is to say that we were not told the orientation of the upper triangle relative to the lower one and yet expected to solve the puzzle. So we conclude the orientation does not affect the answer and we can choose it to be anything we like. Very well, let's position the upper triangle directly above the lower, like this. 

Now it's self-evident the angle sought is 30 degrees and from the argument just given we see it must have this value irrespective of the orientation of the upper triangle. Another example where the absence of what seems a crucial piece of information turns out to be the key that unlocks the puzzle!

Here are two views of a puzzle object I bought on Amazon.

How can this be? I promise you there is only one object and the viewpoint is all that has changed.

According to Wolfram MathWorld "Any triangle can be positioned such that its shadow under an orthogonal projection is equilateral. "

You can try this for yourself.

I cut a triangle out of printer paper and taped it to a cocktail stick. Then we put a sheet of paper under the kitchen light and a friend experimented with positioning the triangle until we got a shadow that was a pretty fair equilateral triangle. Here's the photo.

A is the paper triangle (the shadow behind is my friend's hand) and B is the triangle's shadow, which you can see is more or less equilateral. It took two or three minutes of trial and error to get the position right but we were pleased with the eventual result!

In Greek mythology the Muses* were the daughters of Zeus and Mnemosyne, whose name is connect with mnēmē "memory", which gives us mnemonic. This probably, although not all authorities agree, comes from a PIE root  *men-, which is to do with thinking and mental (!) capacity in general, and is connected with many words, some quite surprising, for example Ahura Mazda, comment, mandarin, mantra, monster and mosaic — and of course Muse.

*The Muses

Wikipedia gives the following list of Muses and the art or science associated with each, although as you would different authors, classical and later, did not always agree with this list and even with how many the Muses were.

Calliope (epic poetry)
Clio (history)
Polyhymnia (hymn and mime)
Euterpe (flute)
Terpsichore (chorus and dance)
Erato (lyric choral poetry)
Melpomene (tragedy)
Thalia (light verse and comedy)
Urania (astronomy and astrology)

Acknowledgement

Copilot generated the image 20 Sep 2025

A friend in Devon sent this photo yesterday.

After some Googling I found it is a Portuguese Man o' War and quickly messaged my friend warning her to give it a wide berth, because the tentacles you can se in the photo are highly venomous. and in fact the Portuguese Man o' War is seriously dangerous.

It has been quite windy and stormy there and the animal must unfortunately have been blown off course in the Atlantic and become stranded on the beach.

Actually, from my reading, I found it is not an animal but a siphonophore, a whole colony of animals referred to as zooids, each multicellular, and all genetically identical, but having specialised into different roles. One (or maybe some, I am not sure) forms a gas-filled bladder which acts as float for the whole colony. Other zooids form the venous tentacle which incapacitate other animals and presumably draw them back to the colony. A third type of zooid there digest them, and presumably share the nutrients, I don't know how.

Finally a fourth zooid is responsible for sexual reproduction.

In life the colony might have look a bit like my sketch.

There are many other surprising facts about this strange life-form. One I particularly like is the small fish that has evolved immunity to the venom and now lives with impunity amongst the tentacles feeding on any left-overs.

Problem 1
I found on this puzzle 'Watermelon Paradox' on the Mind Your Decisions YouTube channel which cites the original source as the 2010 Indian KVPY exam, but it is probably a classic problem; I found 3,570 Google hits for it. It runs like this (my paraphrase)

I have 100 kg of watermelons which are initially 99% water. After a few days they have lost some moisture and are now only 98% water. How much do they weigh at this point?

Problem 2
This one is very common, with 169,000 Google hits, and I don't know where it originated. It popped up earlier this year and the Mirror wrote of it 'Incredibly hard 1% Club question leaves Brits arguing over the answer' ^[1]. Here is how this one goes (my paraphrase again).

In a room are 100 people, of whom 99% are left-handed. How many left-handers must leave the room to make the proportion of left-handers drop to 98%?

These two problems are the same really, only the way they are framed is different. In each there is an answer we might immediately jump to, but which on reflection turns out wrong. This is quite a good example of the System 1 thinking, as described in the 2011 book Thinking Fast and Slow by Kahneman ^[2]. The book contrasts two modes of thinking

System 1 is fast and intuitive, a snap answer that takes little effort to arrive at and which we don't really have to think about.

System 2 is slow and takes effort, we have to think about how to solve the problem and work the answer out methodically.

Both these systems have their place.  We often must make quick judgements and much of daily life is governed by small familiar decisions taken unconsciously and here System 1 is needed. But other decisions may concern problems that are complex and unfamiliar, with answers that are unintuitive. Here System 2 is required.

Both the problems I have given above trick us into System 1 thinking, so we are tempted to say 99 kg, or 1 person. The correct answers are highly unintuitive; 50 kg and 50 people, which is quite surprising. Especially with Problem 2 I find the answer hard to believe even though I know it's right.

Here is my System 2 analysis of the two problems side by side, so you can see how they parallel one another.

[1] https://www.mirror.co.uk/news/weird-news/incredibly-hard-1-club-question-34387903

[2] https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow

Wiktionary defines an antiproverb as: " A humorous adaptation of one or more existing proverbs."

There are many forms but a common one starts with one proverb, then switches in midstream to another. For example, the daftly incongruous

"Every dog has a silver lining"

Or the sardonic

"No news is the mother of invention."

Here are more examples from [1]

Don’t count your chickens in midstream
You can lead a horse to water, but you can’t have it both ways.
Too many cooks are better than one.
An apple a day is worth two in the bush.

The word antiproverb was coined by Wolfgang Mieder and there is quite a literature about antiproverbs ^[2][3]. People who study proverbs are paremiologists, a word new to me but paremiology is in the OED and attested from 1861, derived from Latin paroemia, Greek παροιμία.

I thought I would try to generate some antiproverbs of my own, so I got a list of just under 1,000 proverbs and generated many random pairs, looking for good combinations. It turned out harder than I was expecting, but here are some that I feel show definite promise

An apple a day is better than no bread.
Don't count your chickens while the sun shines.
Many a true word is sauce for the gander.
Don’t change horses till the fat lady sings.
It’s an ill wind that never boils.
It’s easy to be wise after a free lunch.

[1]  https://wordsbybob.wordpress.com/2014/01/20/antiproverbs-say-what/

[2] https://www.degruyterbrill.com/document/doi/10.2478/9783110410167.15/html

[3] https://www.researchgate.net/publication/370487535_ENGLISH_ANTI-PROVERBS_AS_STYLISTIC_DEVICES

More than 2,000 years ago Euclid proved that in any triangle the three lines bisecting the angles of a triangle meet at a point which is the centre of the circle that touches the triangle's sides.

He further proved that the three lines that bisect the triangle's sides at right angles is the centre of a circle passing through the triangles three corners.

These circles, called the incircle (centre incentre) and circumcircle (centre circumcentre) still studied in schools today, for example here is quite a nice animation showing the construction of the circumcircle, from a GCSE revision site.

And if you are interested in Euclid's original proofs here you can see the relevant pages from the oldest know complete copy of Euclid's Elements, with a transcription into a readable form and an English translation. You want Book IV, Elem. 4.4 and 4.5. This website is an astonishing work of scholarship.

All that was just the preamble. Here is a neat fact I stumbled across about a week ago, when I was just doodling triangles. It's nice because it connects the angle bisectors and the perpendicular bisectors.

In a triangle the line bisecting an angle meets the perpendicular bisector of the opposite side at a point (M in the diagram below) that lies on the circumcircle. 

This was new to me but I thought there ought to be quite a simple and accessible proof. But after a bit of head scratching, I couldn't see one, so I thought it must be a standard result, and looked it up. I did find a few proofs, but they were all more complicated than I was hoping (and at least one was wrong). The problem is discussed on Mathematics Stack Exchange but I still didn't find the "obvious" proof I was looking for.

After days of head-scratching I finally had my eureka moment! The proof I was seeking uses the following fact.

In a given circle, any two chords with the same length subtend (i.e.make) equal angles on the circumference. Here's an example:

Now it's easy. Add some chords.

I claim that the line BM that joins B and the point M where the perpendicular bisector of AC meets the circumcircle is the line bisecting angle ABC.

Proof: Any point on the perpendicular bisector of AC is equidistant from A and C. So AM and MC are equal chords, and the angles ABM and MBC they subtend are equal, in other words BM bisects angle ABC.

I saw this problem on the "Mind Your Decisions" YouTube channel and here are my two solutions, I haven't watched the video/

The Problem

In my sketch we have a rectangle of unknown dimensions, a quarter-circle inscribed in the rectangle, a semicircle positioned as shown with its centre on bottom of the rectangle, and a line of length 5 drawn from the corner of the rectangle and tangent to the semicircle. The challenge is to find the area of the rectangle.

At first sight this is impossible, we only have one distance so how can we find the area? 

First Solution

When a problem involves a tangent to a circle or part of one, it almost always uses the fact the a tangent makes a 90 degree angle with the radius at the point of contact; and when a problem involves a right-angled triangle and we are interested in distances, that points to Pythagoras' Theorem.

So here's the diagram again: I have labelled some key points, called the radii of the quarter and semicircle r₁ and r₂ respectively, drawn in the radius to point of contact D, and marked the right angle.

Now we see we have a right-angles triangle with hypotenuse r₁ + r₂ and its other sides 5 and r₂. We can apply Pythagoras and then use some algebra on the resulting equation as follows.

But r₁ and r₁ + r₂ are precisely the height and length of the rectangle and their product is the area of the rectangle, which must therefore be 25. r₁ and r₂ can take different values as long as the satisfy the relationship we ended up with above and the area must always be 25.

Second solution

We are not told the values of r₁ and r₂, so it must not matter as such and we are free to choose them as we like as long as our choice is compatible with the given geometry.

Very well: let's set r₂ = 0. Now the semicircle collapses to a point at B, the rectangle becomes a square, and the tangent degenerates into the line AB, r₂ becomes 5 and the area is (r₂)² = 25.

It's a bit of a mystery.

In most of the languages of Europe the word for dog comes from an ancient PIE root which was something like kwon- and which appears in e.g. Latin canis, Ancient Greek kyon and Welsh corgi. In Germanic languages the k sound became h, so we get Modern German Hund and Old English hund (another example of this consonant shift is Greek kardia versus English heart, German Hertz).

For today 'hound' is reserved for dogs (or people) that hunt or track something bloodhound, newshound, or breeds of hunting dogs wolfhound, or used metaphorically, or perhaps in a jokey way. At some point in late Old English a word docga (possibly referring to a spacial breed of strong or powerful animal) emerged from completely unknown origins. During the Middle English period dog displaced hound as the standard word for members of the genus Canis and the meaning of hound narrowed to the more restricted sense it is used in today.

To me hound has more portentous feel than dog: "The Dog of the Baskervilles" wouldn't really worked.

A palindrome is of course a word (e.g. tattarrattat, a knock at the door) or phrase (e.g. Norma is as selfless as I am Ron, my favourite) that reads the same forwards and backwards,.

But have you heard of the anadrome? An anadrome is a word the taken backwards gives a different, but perfectly good, word. The longest example in English is, as far as anyone knows, the 8-letter stressed and desserts.

I wrote a short program that searched a public-domain word list and found several 7-letter anadromes:

dessert
reviver
reifier
stinker
stellas
deifier
deified
deliver
reviled
rewarder
halalah
reified
sallets
reknits
stressed
sememes
redrawer
tressed

I rather like stinker and reknits.

If we consider shorter words there are many more anadromes: I found 397 of 2 or more letters in a list of 113809 words.

There are also some that have been made up: the unit of electrical resistance is the ohm and as far back as 1888 someone coined mho as a unit for the reciprocal of resistance, conductance. It's probably not official but I think it's quite common, I was certainly familiar with it. Another nice example I found on Wikipedia is tink, which means unknit (geddit?). This goes rather nicely with stinker and reknits, I feel.