OU blog

Personal Blogs

Richard Walker

What I'm Reading

Visible to anyone in the world
Edited by Richard Walker, Friday, 6 Dec 2024, 23:47

More dipping into really. It'sĀ Mathematical Puzzles: Revised Edition, by Peter Winkler. It's rather good, but I only stumbled across it by accident. Somehow (I can't remember how) I read a Guardian newspaper puzzle column and it contained this gem from Winler's book:

Find all the ways to arrange four points so that only two distances occur between any two of them.

There are six distances among four points and in general they are all different, as seen the left-hand sketch below. The sketch on the right however shows a configuration in which only two different distance are represented; the side of the square are one length, and the diagonals the other. This is one solution to our puzzle then, but it is not the only one. How many others can you find? In Winkler's words "There are more of them that you probably think".


If you want to see the full solution the Guardian column provides a link to it.

This got me thinking about related problems.Ā  With five points you can still keep to two distances but there is only one way (I think.) What about six points? Well, three different distances is the minimum possible number, but how many ways can it be done in? So far I have found four but I haven't tried yet to prove these are the only ones. Here are four I've found, with descriptions.

Permalink
Share post
Richard Walker

Winter Rose Haiku

Visible to anyone in the world

Winter Rose


One last rose
Alone in the winter garden.
So red it hurts.






Permalink
Share post
Richard Walker

Word of the Day ā€“ Grawlix

Visible to anyone in the world

As Merriam-Webster puts it [1]

What the #@*% Is a ā€˜Grawlix'?

You will have seen thousands of them, because they are the conventional way to indicate a cartoon character is swearing, like this




There is even an emoji


šŸ¤¬


I've often wondered about them but until this evening had no idea they has a special name. Now I am wiser, thanks to the latest Words Unravelled podcast.

[1]Ā https://www.merriam-webster.com/wordplay/grawlix-symbols-swearing-comic-strips




Ā 

Permalink
Share post
Richard Walker

The fascinating etymology of mistletoe

Visible to anyone in the world

'Mistletoe' comes directly from Old English mistiltan. There's general agreement the -tanĀ element means 'twig' and comes from an old Germanic root with the same meaning. The first element mistle is less straightforward. It's derived from missel, an old word for, well, mistletoe and the origins of this are less clear.

One theory is that it's from a root that meant something like 'urine' or 'feces', from the fact that mistletoe seeds are spread in bird droppings. If so it would be related to 'micturate', i.e. urination.

But many, many plants are spread by bird droppings, so why should mistletoe be singled out in this respect?

Another possibility that mistletoe is named for a related concept of something sticky, and this is plausible, because the berries of mistletoe are indeed very sticky and do adhere to birds' beaks and plumage and get spread that way.

Some support might be offered to this theory from the fact that the Latin word for mistletoe is viscum, which also meant something sticky, and from which we get viscous.

As an aside, the French for mistletoe is gui, and this is one of those words that has endured some surprising changes. It's said that viscum evolved first into wiscum, under Germanic influence, then into gwy, guy and finally gui. (At this point you might be wondering about 'gooey' but sadly there is no connection that we know of.)

The Ancient Greek word wasĀ į¼°Ī¾oĻ‚ and this seems to be cognate withe the Latin viscum; originally it would have started with a 'w' soundĀ ĻĪ¹Ī¾oĻ‚ but theĀ Ļ letter was lost from Greek quite early. There are many other examples of words where the Latin and Greek were similar but the V or W disappeared from the Greek word, for example vicus 'village' andĀ Īæį¼¶ĪŗĪæĻ‚ 'house' (as in ecology and economy)

Permalink
Share post
Richard Walker

Spot the connection joke

Visible to anyone in the world
Edited by Richard Walker, Monday, 2 Dec 2024, 00:30

Two pictures here. Can you see what links the one on the left with the one on the right?


Image: Copilot designer

Permalink 2 comments (latest comment by Richard Walker, Monday, 2 Dec 2024, 10:19)
Share post
Richard Walker

Maths Problem - Sums of Subsets

Visible to anyone in the world

I got this problem from the book Mathematical Puzzles, A Connoisseurs Collection and thought about it in the shower.

Prove that given any 10 distinct numbers between 1 and 100 it is always possible to find two different non-overlapping subsets that have the same sum.Ā 

Here is my solution.

The least a subset can add up to is 0 and the greatest is 91 + 92 +... +100 = 955. The pair of subsets we want to prove exist can't have these values but that doesn't matter.

Imagine we have 956 buckets labelled 0 - 955.Ā 

0Ā  Ā  1Ā  Ā  2Ā  Ā  ...Ā  Ā 955
šŸŖ£Ā  šŸŖ£Ā  šŸŖ£Ā  Ā ...Ā  Ā  Ā šŸŖ£Ā 

Form each of the possible subsets of our 10 numbers, work out its sum, then put that subset in the bucket labelled with the corresponding number. For example the subset {1, 2} goes in the bucket labelled 3.

How may possible subsets are there? For any subset of our 10 numbers a given number is either in that subset or not, so we have 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 210 = 1024 possible subsets.

But 1024 is bigger than the number of buckets, which is only 996. Therefore at least on bucket contains two or more subsets, which must have the same sum.

This pair of subsets are different but might overlap, if so remove the numbers they have in common. This cannot empty either or both of the sets because if it did it would mean they were the same, or had different sums, and neither of those things is true.

Hence we have proved a pair of subsets with equals sums must always exist.Ā 

Permalink
Share post
Richard Walker

Why did the chicken cross the road? - answered in corporate style

Visible to anyone in the world
Edited by Richard Walker, Thursday, 28 Nov 2024, 16:36

I asked Copilot for a long corporate-style answer to the classic Chicken Question and it came up with this. I think it did a pretty good job, probably becauseĀ 

a) there are lots of parodies like this out there for it to draw inspiration from andĀ 

b) itā€™s read millions upon millions of lines of corporate-speak in its training data.

Q. Why did the chicken cross the road?

A.Ā In a strategic manoeuvre designed to optimise its geographical positioning and enhance its market presence, the chicken executed a calculated traversal of the thoroughfare. This initiative was undertaken to leverage the potential opportunities on the opposing side, thereby maximising its resource allocation and achieving a competitive advantage in the avian sector.

The chicken's decision to cross the road was predicated on a comprehensive analysis of the environmental variables and a thorough risk assessment. By engaging in this cross-road venture, the chicken aimed to capitalize on the untapped potential of the adjacent territory, thereby driving growth and fostering innovation within its operational framework.

This bold move was aligned with the chicken's overarching mission to expand its footprint and deliver value to its stakeholders. The crossing was not merely a physical relocation but a strategic realignment designed to enhance the chicken's long-term sustainability and profitability. Through this initiative, the chicken demonstrated its commitment to proactive market engagement and adaptive strategy execution.

In conclusion, the chicken's road-crossing endeavour was a testament to its forward-thinking approach and its unwavering dedication to achieving excellence in its domain.


Permalink
Share post
Richard Walker

Dad Joke Visualised

Visible to anyone in the world

Pyrex of the Caribbean


Image generated by Copilot Designer

Permalink
Share post
Richard Walker

What are frequentatives?

Visible to anyone in the world

Frequentative is a term from grammar, one that I only learned about a couple of days ago while watching the Words Unravelled podcast. A frequentative is a word formed by appending the suffixĀ -le or -erĀ to a verb or noun to express the idea of a repeated action. Here are some examples with -le

spark -> sparkle

prick -> prickle

game -> gamble

daze -> dazzle

crumb -> crumble

fizz -> fizzle

scud -> scuttle

twink (same as wink) -> twinkle

wag-> waggle

This suffix derives from an old Germanic suffix with a similar meaning.

Here are some examples with -er. These are less clear cut sometimes, because the suffix can have other senses.

gleam -> glimmer

slide -> slither

flit -> flitter

The word frequentative sounded fairly modern to me, but in fact it is recorded in the grammatical sense as early as 1534, inĀ Nicholas Udall's Floures for Latine spekynge selected and gathered oute of TerenceĀ 

Permalink Add your comment
Share post
Richard Walker

Tom Swifty

Visible to anyone in the world

ā€œMeet Antony. He used to be a cab driverā€, said Tom exuberantly.

Permalink Add your comment
Share post
Richard Walker

In the Style of Dali

Visible to anyone in the world

The London Eye, by Image Creator (2024)


Permalink Add your comment
Share post
Richard Walker

Tom Swifty

Visible to anyone in the world

ā€œSydney and I have been chums for absolutely agesā€, said Tom frenziedly.

Permalink Add your comment
Share post
Richard Walker

Why is a crouton like a crystal?

Visible to anyone in the world


https://commons.wikimedia.org/wiki/File:Crouton_skibidis_next_to_an_ice_cube.jpg

'Crouton' and 'crystal' both ultimately stem from an ancient root that was something like kreusĀ and meant 'freezing over' or 'crusting'. Crystals were thought in Ancient Greece (not unreasonably) to resemble frost,Ā kryos, and from that was derived krystallos.

Related words are crust, crustacean, a whole series of modern words coined from cryo-, and, surprisingly, custard.

Permalink Add your comment
Share post
Richard Walker

Santa's Cats

Visible to anyone in the world


This image was generated by Copilot and it makes me laugh every time I look at it. I love the eager enthusiam on the cats' faces.

In mythology there are many divinities who ride in chariots drawn by exotic animals, but cats are unusual, although in Norse myth the goddess Freyja is pulled by a pair of cats. These are feisty beasts, as you can see in this illustration.


Credit:Ā Ludwig Pietsch, 1865

Some sources say they are named "Bygul" and "Trejgul" but this is a little suspect, because there are Latin words Biga and Triga which mean respectively a two-horse and a three-horse chariot. So do the names reflect an Old Norse confusion about the meaning of the Latin words? Or perhaps a learned joke by a medieval scribe?

You can read more about bigae, trigae (and even quadrigae) in a fascinating article here.

Permalink 3 comments (latest comment by Richard Walker, Wednesday, 13 Nov 2024, 14:32)
Share post
Richard Walker

Tom Swifty

Visible to anyone in the world

ā€œMy job is studying social insectsā€, said Tom observantly.

Permalink
Share post
Richard Walker

Stinky Pinky

Visible to anyone in the world

I'm surprised I haven't come across this word game before. The way it works is player A thinks of a pair of rhyming words and tells player B a humorous definition for that phrase. B then has to try and guess the words.Ā 

For example, if A says "Young cat's gloves', B needs to guess the answer "Kitten's mittens".Ā 

Another example could be "Jovial parrot" = "Jolly polly". You get the idea. Can you work these out?

  • Solitary pebble.
  • Outstanding farm produce.
  • Melancholy item of footwear.
  • NarcissisticĀ locomotive.
  • Classy marine life.
  • Better cruise ship.
  • Polished detective.
  • Red Square goblin.
  • Amusing currency.
  • More cheerful coal miner.
Permalink
Share post
Richard Walker

Word origin of ā€œbudgetā€

Visible to anyone in the world

From a French word meaning a bag, from Latin and before that from Gaulish. It derives ultimately from a PIE root meaning something like to swell. Many other words have the same root; for example, ball, balloon, bale, bulge and surprisingly, ballot. Apparently, balls were once used for casting votes.

Permalink
Share post
Richard Walker

Not enough information?

Visible to anyone in the world
Edited by Richard Walker, Tuesday, 29 Oct 2024, 14:19

I got this from the Andy Math channel on YouTube. In a. we are asked to find the combined area of the two squares.


We have no information about the relative sizes, so how can we solved this? The YouTube video shows that is possible, by doing the calculations.

However the original statement of the problem already implies the sizes don't matter. So there is no need for calculation. We can just adjust the diagram as shown in b.,so that the big square has side length r, and the small square 0. Now we see the required sum of areas is r2 + 0 = r2. Simples!

Permalink
Share post
Richard Walker

One Liner

Visible to anyone in the world

I thought Iā€™d try shoplifting. But they were too heavy for me.

Permalink
Share post
Richard Walker

Tom Swifty

Visible to anyone in the world

ā€œI donā€™t want social insects living in my ditchā€, said Tom trenchantly.


Permalink
Share post
Richard Walker

Honey Fungus

Visible to anyone in the world

I found this striking fungus in my garden today. I believe it isĀ Armillaria tabescens, the ringless honey fungus,Ā but it's hard to be certain. Rather beautiful.


Permalink
Share post
Richard Walker

One Liner

Visible to anyone in the world

I remember that at school we were taught never to use double negatives. Apparently they're a no-no.

Permalink
Share post
Richard Walker

How Many Esses on "Possesses"?

Visible to anyone in the world

In one of the Inspector Morse books, Lewis repeats to himself the mnemonic "Five 'esses' in 'possess'". I can't trace the quote, but I think Lewis must have needed to use the word in a report and felt nervous in case he spelt it wrongly. (The first book in the series was published in 1975, pre-spellcheckers).Ā 

I thought I would test an AI on this, so I asked Copilot. It said four esses. So I tried to correct it and it said rather patronisingly:

I see where the confusion might be! Letā€™s break it down again:

    • P-O-S-S-E-S-S-E-S

Counting the Sā€™s:

    1. P-O-S-S-E-S-S-E-S
    2. P-O-S-S-E-S-S-E-S
    3. P-O-S-S-E-S-S-E-S
    4. P-O-S-S-E-S-S-E-S

There areĀ fourĀ Sā€™s in ā€œpossesses.ā€ It can be tricky with the double Sā€™s, but I hope this clears it up! šŸ˜Š

So next I asked for some words with five esses in, and now it "knows" the right answer:

Sure! Here are some words that contain five Sā€™s:

    • Assesses
    • Assassins
    • Assessors
    • Possesses
    • Sassiness
    • ...
The reason it got it wrong in the first case is presumably an artefact of the way it parses questions but I don't understand it properly. It's interesting though; these language models can often come up with very useful answers, and they are mosy impressive, but they are also fallible in unexpeted ways.

By the way the OED lists the word possessionlessness, cited first from 1905. This manages a whopping 8 esses and may be the record. Copilot claims the number is seven. And perhaps there is a pattern here: in response toĀ possessionlessnesses (not in the OED but found in an internet search) it says eight. I'll leave you to try some more experiments. I have to stop now because I've used up all the questions I can ask Copilot in this session.



Permalink
Share post
Richard Walker

Tom Swifty

Visible to anyone in the world

"I love sleeping under canvas", said Tom intently.

Permalink
Share post
Richard Walker

Colouring the Plane šŸ”“ šŸ”µ

Visible to anyone in the world

Something that has fascinate me for many years is the fact that, if a disordered systems is larges enough, it will tend to contain pockets of unavoidable regularity. Order will emerge from disorder.

For example, imagine we colour every point of the 2D plane in one of two colours, red or blue say. Then there will inevitably be equilateral triangles all three of whose vertices are the same colour; either all red or all blue. Here's how we can prove this.

Imagine we set out to colour the plane in a way that avoids a monochromatic equilateral triangle. We shall find that this is impossible. Consider the following steps in our attempt.


We start with two points which are coloured red (Step 1).This must be possible; if not, there would no more than one red point in the entire plane, so either all equilateral triangles would be blue monochromatic, or all the ones that avoided the single red point would be. Either way our attempt to avoid monochromatic equilateral triangles would have failed spectacularly.

At Step 2 both the ringed points will have to be coloured blue, otherwise one or both the triangles marked will be red monochromatic.

At Step 3 the ringed point will have to be coloured red, otherwise the triangle marked will be blue monochromatic.

At Step 4Ā the ringed point will have to be coloured blue, otherwise the triangle marked will be red monochromatic.

At Step 5 a monochromatic triangle can no longer be avoided; whether we choose red or blue for the uncoloured point a monochromatic triangle with be created.

Permalink
Share post

This blog might contain posts that are only visible to logged-in users, or where only logged-in users can comment. If you have an account on the system, please log in for full access.

Total visits to this blog: 2368435