Edited by Richard Walker, Thursday 5 March 2026 at 00:15
This classic puzzle popped up on Quora.
In case you haven't seen this before and want to have a think I have left a gap below, so you need to scroll down to reach the main business of the post, which would give the game away if you read it!
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Invertible words are words made entirely out of invertible letters:
B, C, D, E, H, I, K, X
which look exactly the same if you turn them upside-down. (We have to use capital letters, it obviously doesn't work with lowercase.) Seeing the car park puzzle made me wonder what was the longest invertible word I could find. So I found a public domain word list of about 170,000 words and wrote a short Python program to search it for invertible words.
They are quite rare: I only found about 400, so that about 0.24%. The longest dictionary words were all 8 letters
However the word list seems to include a few random place names—perhaps they are ones the compiler of the list had a special fondness for—and so my program also found the 10-letter OKEECHOBEE.
Now this is a town in Florida. It has a lake of about 2.000 km2, serious lake, and about 5,000 citizens. And it has a city limit sign as you approach: here it is upside-down
Unfortunately we have lost the last two letters but you can see the name is indeed the same upside-down. AI Overview has this to say about the name, which is also rather interesting, since it evidently refers to the lake
Hitchiti Indian words oki (water) and chubi (big), translating to "big water".
Edited by Richard Walker, Tuesday 3 March 2026 at 23:10
There's an old joke that goes like this
Library User: 'Have you got a bookmark?'
Librarian: 'Yes thousands, and the name is John'.
Library is our first plant-based word. It came into Middle English as librairie and derives for Latin liber, 'book'. The Romans explained the name as being from the liber tree, whose bark had once been used for writing on.
Tree or not, liber is probably from PIE *lubh-ro- 'peel, leaf', which also shares a relationship with lodge, lobby and loggia, in the sense of an arbour or shelter with a roof of leaves or bark.
In modern Romance languages library has come to mean a bookshop or seller, and English library translates as e.g. French bibliothèque, from ancient Greek βιβλιος (biblios) 'book', after the Phoenician city of Byblos[1], which exported Egyptian papyrus to Greece. Or it could be the word was borrowed from Egyptian into Greek and the city got its Greek name from there. Or maybe the Greeks just garbled the city's old Phoenician name Gebal. Either way, from βιβλιος we get Bible, our second plant-based word.
In Old English a library was called a bochord, 'book hoard' or a bochus, 'bookhouse', both of which were eventually displaced in Middle English.
And that brings us to book, our third (probably) plant-based word. This has cognates in many Germanic languages; Gothic 𐌱𐍉𐌺𐌰(boka)[2], Old Norse bok, modern German, Dutch: Buch, boek. The Germanic root these share is usually linked to the beech tree, because? runes where written on beech board or perhaps because book covers were made of beechwood. Wiktionary gives a possible PIE root which would also be the origin of Latin fagus 'beech' and tree-related words in a variety of IE languages.
But the connection with beech has often been disputed and is far from being universally accepted. The debate has swayed this way and that, and the pendulum has currently swung towards the beech tree explanation. You can read a good blog post about the debate here.
[1] Byblos is probably where the Western alphabet was invented.
[2] This is written in the unique Gothic alphabet, which has Unicode support. If you squint a bit you can see it says 'BOKA' or 'BUKA'.
How Far Apart Are Random Points? An Elegant Expectation
Monday 2 March 2026 at 20:19
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Edited by Richard Walker, Monday 2 March 2026 at 22:34
If and are two random points on the number line between amd what is the average distance between them?
Of course this is not expressed rigorously but I hope it is good enough for the purposes of exploring our problem.*
What we want is the expected value of . the difference between and ignoring sign. In problems like this it's often useful to take and as a coordinate pair, so here I have done this, with the help of Desmos. For each in the unit square the height of the surface corresponds to .
Now to find our average we can do something analogous to how we calculate the mean of a set of numbers, where we add them all up and divide by how many there are of them. In the problem we are looking at we use a continuous version. We cannot add up all the infinite number of values or count them, but what we can do instead is find the volume under the surface and divide it by the area of the unit square, which is .
The volume is made up of two identical pyramids, with a valley between where and are equal and the distance is . The volume of a pyramid is given by . In this case the base of each pyramid is and its height so the volume is .
The combined volume of the two pyramids is therefore and dividing by the are of the unit square which is we find the expected distance between the two random points is .
PS I have seen it argued that the two points divide the unit interval into three segmentsl and because the points are completely random the expected lengths of all three segments (and therefore the distance between the points should by symmetry be . I suppose this is correct but I have a slight feeling of unease. Is it too glib?
* I should have said the points are chosen at random from a uniform distribution.
Edited by Richard Walker, Tuesday 24 February 2026 at 21:12
The alphabet used by the Romans changed over time and for a while "C" was used for both /k/ (as in Kilometre) and /g/ (as in Golf)/.
Spurius Carvilius Ruga, a freed slave who ran a private elementary school in the late 3rd century BCE, is credited with introducing (or perhaps just promoting) the use of a small horizontal stroke to distinguish between the two sounds, so now /g/ would be spelt with a G and /k/ with a C as before. Perhaps he was influenced by his last name being mispronounced as Ruca with a /k/.
This is only a story of course but if true Sp. Carvilius Ruga would have the distinction of making a small but important typographical change that has survived unaltered for more than 2,000 years. Without it we couldn't tell our goats from our coats, our gold from our cold, or our glasses from our classes.
This pattern in quite amazing when you think about it.
It is made up of four different kinds of triangle: one irregular triangle which can be of any shape we like, and three different sizes of equilateral triangle.
Every equilateral triangle is surrounded by three copies of the irregular triangle (top left). The centres of these triangles form an equilateral triangle.
Every copy of the irregular triangle is surrounded by three equilateral triangles, one of each size (bottom left). The centres of these triangles form an equilateral triangle (this is Napoleon's theorem.)
We can extend the patter as far as we like, so it tiles the whole plane.
If we join up all the triangle centres we get a hexagonal lattice, like a honeycomb.
If we join up the centres of all the copies of a particular triangle (all the equilateral triangles of a particular size, or all the copies of the irregular triangle), we get a triangular lattice.
It also has a rich set of symmetries. There are only 17 basic 'wallpaper patterns' in terms of symmetries but I haven't figured out which one this is yet.
Edited by Richard Walker, Sunday 22 February 2026 at 12:04
I knew that virus originally meant simply 'poison' and guessed it came from Latin, but I didn't know its full history. It's attested in late Middle English from about 1400, with the sense of 'pus', from the Latin virus, 'poison' or 'poisonous secretion'.
The PIE root this derives from is *wisu-, or *uisos-, which has cognates in various languages, including English viscous, from Latin viscum, 'sticky flow' or 'sticky substance', which entered the language at about the same time as virus, so these two words are doublets.
A particularly interesting modern descendent of the PIE root is Ancient Greek ιός,ios, 'poison', which has survived into Modern Greek as the word for virus.
There might seem a bit of a gap between virus and ios, but the explanation is that Greek at one time had a 'w' sound, written as digamma ϝ , but this had been lost by the Classical era. There are quite a few Greek words that formerly began with 'w' but later lost it. A fine example is woikos, 'home' which became oikos (think ecology, economy). In Latin the same root gave vicus (think vicinity) and vicus was borrowed into Germanic as wick (think Warwick).
Footnote on digamma. It was so called because it looked like one gamma Γ sitting atop another Γ. The Greek alphabet was carried to Italy and adopted by the Etruscans and from there by the Romans. However the Romans wrote 'v' for both the sounds 'w' and 'v' so they were free to recycle the redundant ϝ as the consonant that has come down to us as ef.
PS while I was looking stuff up on Google 'AI Overview" made a joke. Not a brilliant one, but definitely a joke. The Archaic Greek for poison was *wīsos but as you recall the 'w' sound got dropped and then so did the intervocalic s, leaving just ios.
The AI quipped
If that internal s hadn't vanished, we might be calling viruses "wisos-logy" instead of virology today!
(The language is a bit muddled in the middle portion but you get the idea.)
Napoleon's Theorem says that if we draw an arbitrary triangle and erect equilateral triangles on its sides, the centres of the triangles will form a equilateral triangle. Here is what may the first publication of this fact, in the 1826 edition of The Ladies Diary, which was very strong on mathematics at the time[1]. As you can see the proof is quite verbose and many shorter proof have been found. It's doubtful that Napoleon actually had anything to do with it but presumably somebody attributed it to him and it stuck.
The equilateral triangle formed is DEF in the diagram.
I was musing about Napoleon's Theorem because I really like pretty theorems in plane geometry and it occurred to me that I could reverse the roles of the arbitrary triangle and the equilateral one.
If we draw an arbitrary equilateral triangle and erect equilateral triangles copies of an arbitrary triangle on its sides, the centres of the triangles will form a equilateral triangle.
Here it is
Which is rather neat! It's actually quite easy to see why it must be true, because each copy of the irregular triangle represents a 120° rotation of the previous one about the centre of the original equilateral triangle ABC, the whole figure has 3-fold rotational symmetry, any three corresponding points, for example D' D' and D'', are the vertices of an equliateral triangle.
Now what if we go for a sort of hybrid: start with the figure above and erect equilateral triangles on the two free sides of one copy of the irregular triangle, like this
Now let's draw some centres and join them up
I bet you weren't expecting the rectangle, well, I wasn't anyway! There are no other right angles anywhere in the diagram and it popped out of nowhere.
I haven't proved it mind you. I think I could do it with vectors for example but it would rather messy. I'll see if I can do it just using a Euclid-style proof.
Edited by Richard Walker, Thursday 19 February 2026 at 15:05
This puzzle asked
In a litter of mice 3 are white and the others brown.
If 4 of the mice are chosen at random, the probability that the sample contains all 3 of the white mice exactly equals the probability that it contains none of them.
How many mice are in the litter altogether?
I'll give two solution: the first from an AI, correct and not too hard to follow, but long-winded, the second shorter and more insightful
Solution 1
I asked 'AI Overview and it reasoned essentially as follows (I've abridged its answer but not altered the logic).
Suppose there are in the litter. Then 3 are white and brown We can choose 3 white mice from 3 in 1 way and 1 brown from in ways. So there are ways to include all 3 white mice.
On the other hand we can pick 4 brown mice from in .
We are told the two probabilities are the same and so these two numbers must be equal
Cancelling amd multiplying both sides by 24 we obtain
So we seek three consecutive numbers whose product is 24 and this is satisfied by . So and .
Solution 2
If the 4 mice selected include all three white mice, the mice remaining must include none of the white mice.
Conversely, if the 4 mice selected include none of the white mice, the mice remaining must include all three white mice.
So the situation is symmetrical with respect to the location of the white mice and from the information that the two cases have the same probability we can deduce the two groups must be the same size anf thus there are altogether mice in the litter.
I put this to AI Overview and it gave me a pat on the back!
That is a brilliant and elegant way to solve it!
I actually adapted this from another question I saw, in which one probability was twice the other and I wondered if there were other numbers that gave a nice answer, for example, if the probabilities were equal. So I used the long method to get an equation and found that this would be the case if the litter size were twice the sample size. I thought that was rather neat, but then the penny dropped and I saw it was obvious!
PS This was another chance to extend my LaTeX, I now know how to do binomial coefficients.
Sock has a long and interesting connection with comedy. Socc came into Old English from Latin soccus, which meant a light shoe, or a slipper. The Latin word was probably from Ancient Greek sukkhos (συκχος), presumably also meaning a light shoe, perhaps borrowed from Phrygian or the language of a neighbouring people.
Now it seems that in Ancient Greek theatre actors in comedy wore light sykkhous, in contrast to actors in tragedy, who wore a heavier cothurnus, a kind of boot, translated into modern English as 'buskins'.
Three White Mice — A Problem in Intuitive Probability
Monday 16 February 2026 at 23:05
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In a litter of mice 3 are white and the others brown.
If 4 of the mice are chosen at random, the probability that the sample contains all 3 of the white mice exactly equals the probability that it contains none of them.
Edited by Richard Walker, Sunday 15 February 2026 at 12:47
A few days ago I posted a puzzle I'd seen, to find five circles that would between them pass through all 25 dots in a square 5 x 5 grid. Here is the solution I gave, just found by using trial and error.
That got me interested, so then I tried with a 6 x 6 grid, again using trial and error and exploiting some obvious symmetries and was pleased to find this very nice solution with 6 circles, which has all the eight symmetries of a square.
This got me even more interested, so I did a search to see if I could find any literature on the genera case of circles. I didn't find a lot but I did locate this blog post where the author had written a program to do a brute-force search to the 5 x 5 case and found 84 essentially different configurations that solve the problem.
Then I looked for a trial and error solution to the 7 x 7 case but found it more difficult and didn't make much progress, so I decided to recruit ChatGPT 5.2 as a research collaborator. I first asked if it could locate anything about the general problem but it drew a blank. So then I wrote quite a long and detailed prompt asking it if it could solve the 5 x 5 and 6 x 6 cases.
It quickly wrote and ran a Python program to do a search and after a minute or two came back with the equations of circles solving the 5 x 5 and 6 x 6 cases. I've not yet checked but I think they are essentially the solutions I've shown above. I felt very encouraged so then we moved on to the 7 x 7 case.
At first the best it could do was 9 circles but after tweaking its algorithm it reduced this to 8 and gave me the equations. I typed them into the brilliant and free online application Desmos [2] and here's the result. It works!
Is this the minimum possible? I asked CharGPT and it had a go, writing and running code, but it exhausted its quota of processing cycles without finishing. It then displayed the program and suggested I run it on my local computer, which I did and after a few minutes it finished with the message
'No 7-circle cover exists (CERTIFIED)'
I should say that throughout this conversation ChatGPT had been at pains to stress that I should try to independently verify whatever it told me and of course I had planned to do that anyway. So I next asked, as a test of its algorithm, how many different solutions exist for a 5 x 5 grid. And it said the solution it had given was
Unique
This really surprised me, because I'd found a website that seemed to compute solution to the 5 x 5 and claimed to have found 84. So then I tried to get ChatGPT to re-evauate its conclusion, even sending an image of a different solution and explaining why it was different. But I couldn't shake ChatGPT on this.
So I then I resumed my search for pages that might have relevant content but this time I asked Copilot and it did better that ChatGPT and found a page in the Online Encyclopedia of Integer Sequences. This has lots of information, including the number of circles needed up as far as the 12 x 12 case (which takes 15), the fact that the only cases with unique solutions are 3 x 3 and 8 x 8 (so 5 x 5 is not unique and ChatGPT was indeed wrong about that) and confirmation that 7 x 7 needs 8 circles, as indicated by ChatGPT's program.
And it has a link that would lead to original source of much of this information and would probably answer questions like 'How do we know the 8 x 8 solution is unique? Is by computation or was there some kind of deductive argument?
But unfortunately this link broken and I haven't so far been able the locate the page I want by other routes, so I am paused from the moment.
I started this investigation partly because of the inherent interest of the problem but also because I wanted to explore for myself how useful AIs such as ChatGPT might be as research assistants. In my estimation, very useful indeed, with all the usual caveats about not simply accepting its answers without careful checking. I was very surprised by how well ChatGPT seemed to grasp what I was trying to find out and by its versatility in suggest avenues I might like to have it explore. And I was impressed by how much it got right but also slightly frustrated by its stubbornness when it was wrong.
Stop Press
I just found this page which shows 3 solutions to the 5 x 5 case, and asks 'Are there others?'
This tree, the silver wattle, Acacia dealbata, is a close cousin of the golden wattle, Acacia pycnantha, the national flower of Australia, and both species are native to that continent. Worldwide there are 1,000 species of acacia. But the first plant ever called an acacia is no longer classified as acacia! Let me explain.
The Greek botanist and pharmacologist Dioscorides, in his celebrated work De Materia Medica[1] wrote of an Egyption tree which he called akakia (ακακια). From its seed pod was pressed a fluid that was good for (amongst other things) eye inflammation, shingles, and 'blisters in the mouth' and the tree also produced
... a gum that comes out of this thorn which is astringent and cooling.
and in that you have the etymology of gum. The Greek word Dioscorides uses was kommi (κομμι) and via Latin and Anglo-Norman this made its way into Middle English and ended up as gum, a kind of glue, and then used for chewing gum and bubblegum. Kommi is thought to have been borrowed from Egyptian qmy, 'resin' or 'gum' or 'anointing oil'.
When Linnaeus came to name this tree in 1753 (?) he made it the type species of a genus Acacia, using the Greek name, and called it Acacia nilotica and it was (and still is) widely still used. However more modern botanical research has led to the decision in 2005 to assign it to a different genus and rename itVachellia nilotica, with Acacia only being (mainly) confined to Australian species [2]. (So our tree is a fully accredited acacia.)
This controversial step although scientifically justified, has led to a situation where Dioscorides' ακακια in no longer an acacia, which a imagine would have surprised him.
In this puzzle we are given three doors, one of which hides a car we can win if we are smart. Each door carries a statement and we are given that exactly one of these statements is true. So can we locate the car?
It's tempting to read the statements and start thinking 'If this true, what will follow?' or 'If this s false, what will follow?' and I can feel myself being tugged in that direction. But that usually isn't the best approach with puzzles like this. It's better to imagine putting the car behind each door in turn and see what that does for the truth or falsehood of each of the three statements.
So, suppose the car is behind door 1. Then statement 1 is true (the car is behind door 1) but so is statement 2 true (the car is not behind door 2). But this runs contrary to the fact that only one statement is true. So the car cannot be behind door 1.
Now let's skip to the last door and imagine it is the one that hides the car. Then statement 1 is false (the car is not behind door 1). Statement 2 is true (the car is not behind door 2), and statement 3 is also true (the car is not behind door 1). This again runs contrary to the fact that only one statement is true. So the car cannot be behind door 3.
That leaves door 2. If the car is behind door 2 statement 1 is false (the car is not behind door 1). What about statement 2? Well, that is also false, if the car is in fact behind door 2. And statement 3? Well it's true (the car is not behind door 1), and it's the only statement that is true. So this and only this fits the information we have been given, and the car must be behind door 2.
I think this puzzle has been skilfully crafted, because the car is behind a door that carries a false statement saying it's not behind that door, a neat twist!
Edited by Richard Walker, Tuesday 10 February 2026 at 17:17
Here's a puzzle posted by Presh Talwalkar, which he found posted on Brilliant (although it has been posted in multiple other places as well). I hadn't seen it before and think it's rather nice.
Originally it was about three boxes one of which had a car in it but it put me in mind of the well-known Monty Hall problem (three doors, once of which has a car behind it) so I even though the puzzles are quite different I couldn't resist going with doors.
In the picture I have drawn you see three doors, one of which hides a car. Each door has a statement attached to it and you are told that precisely one of these three statements is true.
Armed with this information can you deduce which is the door hiding the car? If so, you will win tonight's star prize!
'Angus' and the Valkyries. What's the Linguistic Connection?
Saturday 7 February 2026 at 00:32
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Edited by Richard Walker, Monday 9 February 2026 at 14:02
'Angus' and 'Valkyrie' both contain elements derived from the same root as the modern English choose.
Valkyries are female figures in Norse Mythology who hover over battlefields and gather up the souls the slain. carrying them to Valhalla, where they will spend their time alternately feasting and preparing themselves to fight in the last battle, Ragnarök, at which the gods will be defeated and the cosmic order overturned.
Valkyrie literally means 'slain chooser'. The first element valr is an Old Norse word referring to those slain in battle, and the second from Old Norse kyrja, 'chooser', which derives from a PIE root *gues-, whose meaning was 'choose' or 'taste'. It has a host of cognates in different languages, such as Spanish gusto, French goût, English disgust and choose, and Old Irish gus, 'strength', 'excellence', 'choice'.
And that brings us to Angus. The first element here is an-, from the same PIE root as English an, a, one. And you have probably already spotted the second element is gus, 'choice'. So Angus is 'one choice' or 'one excellence'.
It's interesting, and something I didn't know, that the word valkyrie survived into middle English but with meaning of 'sorceress'; the Middle English Compendium records the phrase "Wychez and walkyries' from around 1400.
Word Of The Day — Aphantasia (And Why Is It Always Apples?)
Wednesday 4 February 2026 at 18:17
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Edited by Richard Walker, Thursday 5 February 2026 at 00:03
I'd never met this word until yesterday, and had to look it up. It's a recently introduced word and hasn't made it to the OED yet, but Merriam-Webster defines it as
: the inability to form mental images of real or imaginary people, places, or things
People vary in the extent to which they form mental picture of things and there is a scale, with some of us forming very vivid images and others forming very little and I suppose visualising things in a more abstract way. Think of an apple (don't think of an elephant! We'll come to that later) — what do you 'see' in your mind's eye? If you Google the word and look for images, you will get 700 k hits, many labelled 'aphantasia tests' and typified by this one from the related Wikipedia article
Apple are the most popular choice but birds and horses etc. are also prominent, and there are generally five degrees of ability to form mental pictures.
I never thought about this before and found it quite surprising. I have to say that if I try to rank myself on this scale I come in a firm 4, with only a vague mental picture. It's not that I can't think about an apple, I can, and I can describe its appearance pretty well, and I daresay I could draw a very passable picture of an apple. But I wouldn't be copying some kind of apple picture that's in my head, I'd be creating an image from what I know about apples.
The word aphantasia was coined in a 2015 paper by Zeman et al. [1], the first element meaning 'without' and the second coming from Greek phantasia, 'imagination' or 'appearance'. It's still a rare word (about 1 occurrence per million words on average) but it's Google n-gram shows an exponential growth curve in that short time.
Back to the elephant (had you forgotten it?). There is some other intriguing research [2] about mental imagery that suggests it is harder to visualise oneself moving an imaginary thing if its real-life counterpart would be difficult to move physically. You can try this experiment for yourself. Start a timer on your phone, close your eye and imagine an elephant facing away from you. Visualise yourself rotating the elephant 180 degrees to face you. When you feel confident you have turned the elephant 180 degrees open your eyes and check the time taken for this mental task.
Now repeat the experiment with a smaller animal, a cat say? Do you find a difference. I do (although as I've indicated i don't have very vivid picture of either animal.)
Oh and why are apples used so often? Well my guess is an early version used apples and others have just lazily copied it.
[1] Adam Zeman, et al., "Lives without imagery—congenital aphantasia," Cortex, vol. 73 (December, 2015), pp. 378-80.
Edited by Richard Walker, Wednesday 4 February 2026 at 00:32
The problem below was posted in around 2018 as a mischievous internet meme, '95% of people cannot solve this!'.
Can you find positive whole numbers for a, b, and c?
I missed it at the time and I'm only just catching up. 95% is a bit of an underestimate.
If you'd played around with it a bit and maybe written some code to search you probably wouldn't have got very far, because the numbers in the smallest solution are about 80 digits long! [1]
Python can check the solution for us without blinking.
>>> a = 154476802108746166441951315019919837485664325669565431700026634898253202035277999 >>> b = 36875131794129999827197811565225474825492979968971970996283137471637224634055579 >>> c = 4373612677928697257861252602371390152816537558161613618621437993378423467772036 >>> a/(b+c) + b/(c+a) + c/(a+b) 4.0
But this is nothing! If we change the 4 to 37298 the smallest solution has numbers of 194,911,150 digits [1], which I won't display for obvious reasons.
If we go for 896 we then get trillions of digits, like about 1000,000,000,000+ digits. For those who like statistics, if we printed out three such numbers in 12 point type, on double sided A4 paper, the paper would fill, wait for it... yes, an Olympic Swimming Pool.
For an accessible explanation, not too technical, of how to solve the original question from scratch, with some help from Python, see [2].
For a fuller article that going into a bit more theory, but is very good, see [3].
Edited by Richard Walker, Sunday 1 February 2026 at 21:17
If you love words and love numbers then unusual number words are the tops!
Today's both have interesting histories.
Twain
Mark Twain famously took 'Mark Twain' as his pen name, after hearing it used on Mississippi steamboats to mean a water depth of two fathoms (measured by the second mark on a plumbline).
There is an archaic ring to 'twain', and it is a fossil word, now mainly used in formulaic phrases like 'cleave in twain' and rare, at 0.3 occurrences per million words.
Originally it just meant 'two', but not just two of any old thing, it was two things with masculine grammatical gender. In Old English nouns were masculine, feminine or neuter, as in modern German. Two (and three) were adjectives and had to agree with the thing they were describing, so we had twegen (M), twa (F) and tu (N). (It's actually a bit more complicated, because they also had to agree as to grammatical case, but you get the idea.)
Here are examples, courtesy of AI Overview
Twegen cnitas
Twa cwena
Tu scipu
(A knight was originally a boy (like modern German Knabe) but later changed its meaning.)
The masculine twegen survived as twain, but with a slightly different meaning, and the other forms became modern two. Its survival may have beed aided by its use in the King James' Bible, which intentionally used archaic language.
Thrin
This is a much rarer word at 0.01 occurrences per million words. It is to three what twin is to two, and means something like 'threefold' or it can actually mean one of three children. Although it's from the same Germanic root as three, and ultimately from the same PIE root, it came into English not directly, but via Old Norse þrinnr, 'threefold'.
Edited by Richard Walker, Saturday 31 January 2026 at 23:33
... are getting on well, I really like being able to use professional quality mathematical notation. So far I've only mastered simple algebra and not attempted anything fancy. Here are some examples.
The cube root did throw a bit of a curveball. To get a cube root (or fourth root etc.) you have to use a square root and say it is a a 'cube square' root, like this
Notice I had to use an image because in this editor as soon as you save any LaTeX code gets rendered as mathematical notation and I want to show what the code itself looks like.
The logic is quite straightforward once you get the hang of it. A backslash is like a kind of escape character that takes us into the LaTeX editor and the open bracket ( says we want inline mode.
Then we have another backslash, which signals a command will follow, and the command in this case is sqrt.
Next there is a modifier (if that's what it's called) in square brackets [3], so now we've said its to be a cube root, and then an argument {2} in curly braces, saying what to want it to be the cube root of.
Finally we have to jump back into the normal editor, and we do that with a last backslash and a closing round bracket.
When you enter all this it looks like the image displayed earlier but then when you click the 'Save changes' button and what you have written is posted it comes out like this.
All this is not too hard with a little practice and there are plenty of places you can look up the commands you want. The main difficulty I've found is that with so many brackets it's easy to miss one. Then when you save the changes you get a reasonably helpful error message but then you have to go back and edit the post.
Edited by Richard Walker, Friday 30 January 2026 at 22:18
Here is my solution to the Japanese Temple Problem (Sangaku) I posted here a couple of days ago.
The problem ask for the radius of the small green circle in Figure 1 below, assuming the radius of the largest circle is 1 unit.
Figure 1. Sangaku problem
Figure 2 shows the construction, which is followed by the explanation.
Figure 2. The construction
Explanation
O and P are respectively the points at which the red circle is tangent to the blue semicircle and its diameter MN. Q is the centre of the red circle.
Draw a line bisecting PQ at right angles. The distance CP is and the perpendicular distance between any point on the bisector and the diameter MN must therefore also be .
Next draw a circle with centre Q and radius to intersect the bisector of PQ at E. Now draw the green circle with centre E and radius .
To show this is the required circle we need to show it is tangent to the diameter, the red circle and the semicircle.
Because the radius of the green circle is \(\frac{1}{4\}) and that is also the distance between the bisector and the diameter, the green circle and the diameter must be tangent at H.
Because the distances QE and FE are and by the construction and QF by assumption, QE = QF + FE and so the red and green circles are tangent at F.
Because CE is the perpendicular bisector of PQ, PE= QE = , PG must pass through E, and PG = OE + EG = + = . G therefore lies on the circumferences of the blue and green circles and must be the point at which the are tangent.
The answer to the problem is therefore , a very neat result.
Edited by Richard Walker, Thursday 29 January 2026 at 23:13
Perhaps surprisingly, given any sequence of digits whatsoever, there are infinitely many prime numbers that start off with that sequence. For example, take 2026 as the sequence. Here are 19 primes beginning 2026
There are lots of others but I've chosen ones that grow by a order of magnitude each time because I think it suggests an argument that we can go on finding similar primes for ever.
Consider the
9 integers 20261 - 20269
99 integers 202601 - 202699
999 integers 2026001 - 2026999
and so on. Now a good estimate of the average gaps between consecutive primes near a given is known to be (the natural logarithm of . If we divide the sizes of the intervals, 9, 99, 999... by the logarithms of 20265, 202650, 2026500... (the midpoints of the ranges) that should give the approximate number of primes we can expect to find in each range. This gives
20261 - 20269 estimate 1 actual 1
202601 - 202699 estimate 8 actual 8
2026001 - 2026999 estimate 69 actual 72
The estimates are pretty good! The numbers are actually growing, which more than supports the contention that primes starting with 2026 are infinite in number. This can actually be proved properly; my heuristic above seems plausible but is not an actual proof.
Edited by Richard Walker, Friday 30 January 2026 at 13:17
Sangaku were geometrical puzzles from the 18th, 19th and early 20th centuries, painted on wooded tablets and hung in Japanese temples. Here is a problem I came across which is either a Sangaku or inspired by that tradition. It is very simple to state.
Inside a circle another smaller circle is drawn which is tangent to the bigger circle and to a diameter of the bigger circle.
An even smaller circle is then drawn which is tangent to the diameter and to both the other circles, as shown in Figure 1.
Figure 1. The green circle is tangent to the red circle, the diameter and the enclosing blue circle.
What is the radius of the smallest circle, as a fraction of the radius of the biggest circle?
Can you see how to construct the smallest circle using straightedge (i.e. a ruler with no makings on it) and compasses? If you can it should help you answer the first question.
I had a lot of fun solving this problem which turns out to have a really nice answer. I'll post my solution, which I am pretty comfident is correct, at the end of the week.
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