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 includeĀ 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 aĀ 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 binomialĀ coefficients.
Sock has a long and interesting connection with comedy.Ā Socc came into Old English from Latin soccus, which meant a light shoe, or a slipper. The Latin word was probably from Ancient GreekĀ sukkhos (ĻĻ ĪŗĻĪæĻ), 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 ā 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 theĀ 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 forĀ a 5 x 5 grid. And it said the solution it had given wasĀ
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?Ā
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 calledĀ akakia (Ī±ĪŗĪ±ĪŗĪ¹Ī±).Ā 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 producedĀ
... a gum that comes out of this thorn which is astringent and cooling.Ā
and in that you have the etymology of gum. The Greek word Dioscorides uses was kommi (ĪŗĪæĪ¼Ī¼Ī¹) 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 genusĀ 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' Ī±ĪŗĪ±ĪŗĪ¹Ī± in no longer an acacia, which a imagine would have surprised him.
In this puzzleĀ 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,Ā Ragnarƶk, at which the gods will be defeated and the cosmic order overturned.
Valkyrie literally means 'slain chooser'. The first elementĀ 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 goĆ»t, English disgust and choose, and Old IrishĀ 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'.Ā 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 ā 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
:Ā 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) ā 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 beforeĀ 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.Ā
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 haveĀ very vivid picture of either animal.)Ā Ā
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 imageryācongenital 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!'.
Ā Ā
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 masculineĀ twegen survived as twain, but with a slightly different meaning, and the other forms became modernĀ 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 NorseĀ Ć¾rinnr, '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.Ā
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 aĀ 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 givesĀ
Ā
20261 - 20269 estimate 1 actual 1
202601 - 202699 estimate 8 actual 8
2026001 - 2026999 estimate 69 actual 72
Ā
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 provedĀ properly; my heuristic above seems plausible but is not an actual proof.Ā
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.Ā
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.
Edited by Richard Walker, Sunday 25 January 2026 at 20:51
Its claim to fame
and (unless you count ) this is the only square number that is a sum of squares of consecutive numbers starting at 1. This means that if you make a square pyramid by piling up cannonballs the only possible number of cannonballs is 4,900.
Proving 4900 is the only solution is not at all easy. The first attempts were in the late 19th century and different proofs were published over a span of more that 100 years. See https://en.wikipedia.org/wiki/Cannonball_problem
Edited by Richard Walker, Saturday 24 January 2026 at 22:57
I just stumbled on this word while looking up something else.
It's a rhetorical device, a figure of speech. Here it is being used in the Bible.
"I will lift up mine eyes unto the hills, from whence cometh my help.
My help cometh from the Lord, which made heaven and earth."
And here by Barack Obama.
"...this is a country that can make you proud.
Proud of our people, proud of our history..."
And Bob Dylan.
"If You Gotta Go, Go Now"
You can see how it is structured, and it does add a lot of eloquence to these sayings. The word means something like "folded back" in the original Ancient Greek. In modern Greek double is still diplo (I've often said this in bars) and the ana prefix meant something like "up" or "back" on Ancient Greek.
A related work is diplomat. This started out as Ancient Greek diploma, meaning a folded paper, then a licence, then via Latin and probably French it reached English, having acquired the sense of any kind of official document, and then the meaning widened to the modern sense of an accredited go-between, then to diplomacy as what such persons did, and then to diplomatic on the sense of tactful.
And of course we still give out diplomas in the original sense.
Edited by Richard Walker, Friday 23 January 2026 at 21:55
I asked here what ratio between the height of a pyramid and the length of its sides would give the greatest volume for a given surface area. Below I will give two solutions, the first I think quite insightful and intuitive, and doesn't need much maths at all, but you have to take a particular fact on trust. The second is more fully worked out but does need some maths.
As well as the side length and height I've added the slant height to the sketch, because we shall use it later, and because it is often referred to in discussion about the proportions of pyramids. The Great Pyramid of Giza originally stood about 147 m tall and its side length at the base was about 230 m. If we work out the slant height and compute its ratio to half the base length it comes to about 1.623, which is sometimes said to be strikingly close to the Golden Ration.
And now: the solution to the puzzle. The height should be <drum roll>
times the side length. It's worth noting that this makes the slant height to half side length ratio , so Pharaoh's new pyramid will be a bit pointier than that at Giza with its ratio of 1.623.
Explanation 1 (intuitive)
Imagine the pyramid is just the top half of a solid whose other half is an inverted copy of the pyramid. What would it look like? Well, it would have eight faces and it's not hard to see it would be an octahedron.
If you like you an imagine the bottom half being buried in the sand!
Our goal is to maximise the volume for a given surface and the octahedron that does that is the regular octahedron. It's also not too difficult to workout that if the octahedrons sides have a length of its height will be the diameter of a square of side length , which is and the height of the top half one half of that, which is .
This is a very pleasing explanation which feels intuitively correctĀ but of course I have just claimed that the regular octagon is optimal without proof. However I'm not going to attempt a proof here, because I think making it watertight would be quite lengthy. If I can find or come up with a nice simple proof I'll write about it a separate post.
Explanation 2 (mathy)
Let the combined area of the four faces be , the slant height , the vertical height , as shown. Let the volume of the pyramid be .
Then firstly, using the formula for the area of a triangle and Pythagoras gives and respectively.
To avoid square roots later we square the first equation so we have . By substituting for and making the subject we obtain
Next the formula for the volume of a pyramid tells us that and it is convenien to square this also, giving . Substituting the expression for found earlier we obtain
We want to maximise rather the but they will both be maximised at the same value of . So we differentiate this expression and find the non-zero root. The derivative is
which is zero when . Plugging this into our expression for and tidying up we get .
Finally we get and that is the ratio of the optimal height to the side length.
PS This has bee a great project to practice my LaTeX!
Pharaoh has decided he doesn't like his pyramid and wants a new one, following the conventional square-based design, but faced with with brilliant white marble that will sparkle in the sunlight. Trouble is, brilliant white marble comes at a hefty prices, and just at the moment the palace coffers are running a bit low.
So he calls in the palace mathematicians and asks them how to make the volume of his new pyramid as large possible for a given amount of marble. What is the optimal ration between the pyramids height H and the base length L of its sides?
This morning I watched a video from YouTuber Anton Petrov ('Why Did Consciousness Evolve?'). This was primarily about birds but at one point he mentioned there was research suggesting that even ants might be self-aware.Ā
I found this idea quite startling, so I did some reading and eventually managed to track down a couple of relevant referencesĀ [1][2]. There is indeed some research, although the findings are far from conclusive and it isn't really clear what if anything the study demonstrates. But it is certainly a very intriguing possibility. The research used the 'mirror test', a well known approach you may have heard of.[3]
The test is design to see if an animal, when looking in a mirror, is somehow aware that what they see is 'them'. Thus of course does not imply consciousness but it does point to some kind og bodily self-awareness.
The usual approach is to put a small coloured marker on the animal's head, in a position that means the animal can't see it directly but will be able to when looking in a mirror. The the animal is then placed in a space where there is a mirror and the experimenter watches for sign the animal has seen itself in the mirror, noticed the marker, and responded, by grooming itself or seeming to be trying to remove the marker. If so, it is argued that it must have recognised that what it saw in the mirror was itself.
Naturally the various factors in the experiment must be systematically varied. Some animals are marked but not put with mirror, some are put with mirrors but not marked, some neither of those, some marked but with markers that are indistinguishable from the animal's natural colouration, and so on.
This experiment has a long and interesting history and been done with many kinds of animal[4]. It seems generally accepted a demonstrating self awareness of some sort in a very wide range of animals.
But the study on ants has been controversial, because of factors such as sample size and poor controls. Controls are inevitably a problem with this sort of experimental setup. For example how can you make it double-blind? And then the observer may be biassed in favour of a positive result, but what counts as positive is just a matter of interpretation. And perhaps mirrors and stickers do not alone change ant behaviour but the combination does, but not because the ant recognises itself, but for some other reason.Ā
So although this is a very intriguing study I think we should be cautious about jumping to conclusions. But it is part of a larger body of work which I think makes it clear that, in the past, we often have massively underestimated the cognitive abilities of many of our fellow creatures.
Edited by Richard Walker, Sunday 18 January 2026 at 22:53
In the words of the prophet Isaiah: ā To the captives, āCome out,ā and to those in darkness, āBe freeā. āĀ
This is from a speech made by George Bush in 2003 and it's notable because of the nested quotes. Bush was quoting Isiah and the latter was quoting what I assume to be the words of God.
Bush's words were quoted in a Guardian article by George Monbiot and perhaps you can see where this is heading.Ā
Another writer 'Mr Smallweed' (actually Davis McKie) picked up on this and recognised its potential. He wrote a humorous piece in which he quoted himself explaining to Mrs Smallweed what what George Monbiot had written.
So at this point Smallweed is quoting himself quoting what Monbiot had written, in which he quoted Bush who had quoted Isiah who had quoted the words of God. The paragraph ends with a flurry of punctuationĀ
ā ā ā ā ā
I hope you are keeping up, but we are not finished yet. It seems Mrs Smallweed met a friend, and Mr Smallweed quotes what she said to her friend, quoting what Mr Smallweed had told her earler.Ā So now
Mr Smallweed is quoting Mrs Smallweed quoting him quoting Monbiot quoting Bush quoting Isiah quoting God
and the triumphant end to the piece isĀ
āTo the captives, ācome outā, and to those in darkness, ābe freeā. ā ā ā ā ā
Ā
You can read the full story in this Guardian piece by David Marsh.
What Links: A Farm, A Throne, The Firmament, and a King of Ancient Persia?
Saturday 17 January 2026 at 00:52
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Edited by Richard Walker, Saturday 17 January 2026 at 00:56
The first three words have reached us via Old French and trace back ultimately to aĀ Proto-Indo-European (PIE) root *dher-, "support firmly".Ā
Farm comes from Old French ferme. This originally meant a rent but came to mean the land being rented and then just the land. It comes from Medieval Latin firma, a fixed payment, which derives from Latin firmus, "stable, fixed", and ultimately from the root *dher-.
Throne is from Old French,Ā trone, and this comes from Latin thronus, from Greek thronos, "throne", and this is thought to also be fromĀ *dher-.
Firmament is from Old French firmament, from Latin firmamentum, "firmament", again fromĀ firmus but with a sense of a strong support, and thence to the sense of the roof of the heavens
this brave o'er hanging firmament, this majestical roof, fretted with golden fire
as Hamlet calls it.
I stumbled across all this because I heard about someone named Darius, which I remembered was the name of more than one king of Ancient Persia, and I wondered what it meant. In Old Persian it wasĀ Darayavaus, with the first element once more from *dher-and the second meaning something like the "the good".
I looked the name Darius up and found the PIE root and then looked that up on https://www.etymonline.com/ and that's where I read about these really surprisingĀ connections. It had never occurred to me that throne and farm might be related.
Five Bridges and a Storm ā A Perplexing Probability Puzzle
Wednesday 14 January 2026 at 23:21
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Edited by Richard Walker, Thursday 15 January 2026 at 23:18
Here's a probability question I found in the marvellous problem collection Cut the Knot.Ā
In Figure 1 (a) we see two islands in a river and five bridges joining the islands to each bank and to one another.
Unfortunately a recent storm has caused damaged and each bridge has a 0.5 probability of having been swept away. What is the probability that a traveller can still cross the river?
This was harder than I was expecting and I found it quite confusing to think about. It did remind me though of a celebrated problem, theĀ seven bridges of Kƶnigsberg, whichĀ prompted to dispense with some of the details and draw Fig. 1 (b), which captures the structure, and then after a bit of head-scratch I reasoned as follows.Ā
Each bridge is either intact or has been swept away, with an equal 0.5 probability of each. So there are 2x2x2x2x2 = 32 equally likely possibilities, not too many. So I could systematically enumerate the ones that allowed a crossing, which I did, and then just count them and divide by 32. Here they are (Figure 2), where the thick lines represent bridges that have survived the storm. I hope I have included them all!
So the probability is . I looked up the answer and sure enough 0.5 is the probability. But the problem setter had a much cleverer way of finding the answer!
Suppose a steamboat is coming along the river.
The dotted lines show the possible channels the boat might follow, if the corresponding bridge is has been swept away. Moreover if we draw a diagram of these channels and their interconnection it turns to be structurally identical to Figure 1 (b), and the boat will be able to pass along each channel with probability 0.5, because if the probability a bridge is intact is 0.5, the probability it has been swept away is also 0.5.
So probability(boat can pass along the river) = probability(a traveller can cross over the river)
And now for the masterstroke! If the boat can pass there must be an unbroken path along the river and the traveller is cut off. If the traveller can cross there must be an unbroken path across the river and the boat is cut off. So the two probabilities above must add up to 1 and hence both are 0.5.
This is a very pretty solution. It seems to have originated in a book Bas [1], as an example where arguing from symmetry gives a correct probability, which in some cases it may not.
And never forget the old malaprop: Don't burn your bridges before they're hatched.
[1] Bas C. van Fraassen,Ā Laws and Symmetry, Oxford University Press, 1989
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