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Richard Walker

A Dotty Puzzle for the New Year

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On this 10 x 10 square grid I have drawn some rectangles with their corners on points of the grid and their sides parallel to the sides of the square. The colours have no particular significance I just used them to make the picture more interesting.

There are obviously many other rectangles that can be drawn on this grid. Can you work out how many there are altogether? Answer on Tuesday.

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Richard Walker

Meet the Isobaths

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I didn't know this word but it was the last answer in yesterday's crossword. The cryptic clue was 

Line showing equal water depth is over tub (7)

This baffled me but a friend suggested the word play is IS + O + BATH = ISOBATH (tub is 'bath', get it?) and there must therefor be such a word. Sure enough when I looked up isobath it means a line of equal water depth. Merriam-Webster gives the definition as:

an imaginary line or a line on a map or chart that connects all points having the same depth below a water surface

As well as this meaning the OED provides a second definition, for a special kind of inkwell:

Trade-name for an inkstand with a float so contrived as to keep the ink in the dipping-well at a constant level

Ingenious but I suppose largely obsolete.

Since water is involved is there any connection with baths? Well no, isobath is from Latin and Greek. iso means 'the same', in Latin/Greek as in isobar or isotherm, and the bath element means 'deep' in Greek, as in bathysphere. Bath on the other hand is an English word of Germanic descent and comes from a root that originally had connections with warming, so applying the word bath to hot springs is very appropriate. German Bad meaning bath is essentially the same word.

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Richard Walker

Tom Swifty

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"Your decision to lower the pitch of the doorbell by a semitone was pure genius", said Tom flatteringly.

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Richard Walker

Wind Haiku

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New Year's wind

Rattling this old house.

Maybe it's time for a shake up.

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Richard Walker

Can you guess the phrase?

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Edited by Richard Walker, Friday, 3 Jan 2025, 01:44

Merriam-Webster defines it as: "an unrealistic enterprise or prospect of prosperity". What is the expression? The picture above is a clue.

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Richard Walker

Limerick — Homage to E. L.

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Richard Walker

An old puzzle and a new surprise

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Edited by Richard Walker, Saturday, 28 Dec 2024, 00:31

—You've probably seen this puzzle, which goes back to at least 1913 and was not new then.

Three utility companies, Gas, Water and Electricity, want to each connect a supply to three houses, Nos. 1 and 2 and 3 Topology View. For technical reasons they do not want anyone's connect to cross another's. The sketch below shows a partial solution, in which 8 of the 9 connections required have been made without any crossings. 

But making the last connection from E to No. 1 seems impossible without a crossing, and indeed it is. A particularly elegant proof is given in this Wikipedia article.

But all of this assumes we are working on a flat plane. What if we change the setting so that suppliers and houses are on a giant donut aka as a torus, as shown below?


Well now we are on a torus we can take a path for E to No.1 that goes round the back of the torus, up through the hole in the middle, and connects to No.1, as indicated by the dotted line in the next sketch.


This configuration of two sets of three points (aka vertices), with lines (aka edges) between point in the first set and every point in the second is called the complete K3,3 which is reasonably self-explanatory. It is one of two "special" configurations, the other being K5 . K5 also cannot be drawn in the plane without a crossing, as we see in the next sketch.


Although I have labelled it K5 that is not quite accurate, because it isn't complete; it ought to have 10 edges. However similar to K3,3 it is impossible to draw it in the plane without two edges crossing.

These two configurations are special because they let us answer the question "What, not necessarily complete, point and line configurations is it possible to draw in the plane without any crossing edges?" The answer is: exactly those configurations that contain no sub-configurations equivalent (in a fairly simple way but I'll skip the detail) to K3,3 or K5.

This is amazing, if you think about it: there are just these two forbidden sub-configurations and if they are avoided any point and line configuration whatsoever of the kinds we have been looking at can be drawn,"embedded", in the plane with no crossings.

Back to the torus. We have seen that K3,3 can be embedded in the surface of a torus, and so can K5 it turns out. In fact we can embed K4,4 (so we could fit in broadband too!) and K7. So we might ask: is there some set of forbidden configurations that let us say what can and cannot be embedded in the surface of a torus, similarly to the way we answered the question for the plane?

The answer is yes there is, but non-one can say yet how many forbidden sub-configurations there are. All that is known at present is that they number at least 17,523. That is the surprise alluded to in the title of this post. When I read that going from the plane to the torus involved a jump from 2 to 17,523 I was staggered!

And of course a torus has only one hole, but we could have 2, 3, 4,... holes and so on, so whatever happens there? It's mind-boggling!


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Richard Walker

Mental arithmetic trick

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Edited by Richard Walker, Thursday, 26 Dec 2024, 16:10

Soon it will be 2025 and that reminded me of an arithmetic trick that used to impress my students, when I was a maths teacher.

Suppose you want to square a number ending in 5, for example 85. Take the first digit, 8, and add 1 to it to get 9.

Now work out 8 x 9 = 72, then put 25 (it's always 25) at the end, giving 7225

And sure enough 85 x 85 is 7225.

This works with any number that ends in 5, for example 1152 is 13225, although doing the calculation in your head obviously gets progressively harder.

What has this to do with 2025? Well 2025 is a square you see; using the method above we find 452 = 2025. This may seem unremarkable, after all, aren't there plenty of square numbers (infinitely many in fact)?

But they get further and further apart the further we travel along the number line – so for example by the time we reach a million only (1,000/1,000,000) x 100% = 0.001% of numbers up to that point are squares. In the long run the proportion of squares gets closer and closer to zero.

And the next square year after 2025 will be 462 = 2116, another 91 years off.

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Richard Walker

Xmas Knock-Knock #7

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Knock-Knock!

Who’s there?

Anna.

Anna? Anna who?

Anna bells were ringing out
for Christmas Day.


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Richard Walker

Not actually reindeer but the kids were thrilled 🦌

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Some friends' children were walking to church yesterday and were enchanted to see Santa's deer crossing the skyline.


You can see some of them have well-developed antlers and my guess is they were fallow deer, which are common round here, but hard to be sure from so far off.



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Richard Walker

Xmas Knock-Knock #6

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Edited by Richard Walker, Monday, 23 Dec 2024, 23:57

Knock-Knock!

Who’s there?

Lars.

Lars? Lars who?

Lars Christmas, I gave you my heart...

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Richard Walker

Xmas Knock-Knock #5

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Knock-Knock!

Who’s there?

Cooking.

Cooking? Cooking who?

Cooking Wenceslas looked out...

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Richard Walker

Xmas Knock-Knock #4

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Knock-Knock!

Who’s there?

Hugh A.

Hugh A. who?

Hugh A. in a manger…

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Richard Walker

Xmas Knock-Knock #3

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Knock-Knock!

Who’s there?

Asquith.

Asquith who?

Asquith gladness, men of old...



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Richard Walker

Xmas Knock-Knock #2

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Knock-Knock!

Who’s there?

Hokum.

Hokum who?

Hokum, all ye faithful!

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Richard Walker

Xmas Knock-Knock #1

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Knock-knock!

Who's there?

Irma.

Irma who?

Irma dreaming of a White Christmas!

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Richard Walker

"Nothing is impossible" - Erm...

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Just saw this motivation video that began

"Nothing is impossible"

But if that is true we will never be able to find anything impossible. So finding something impossible will be impossible.


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Richard Walker

Etymology of 'Xmas'

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We were taught in school that the 'X' here is being used to represent first letter of Χριστός, Christós, the Greek for Christ, and so Xmas is just short for Christmas. This explanation certainly seems to be true and until tonight I never gave much thought to how long the abbreviation has been in use or why it was adopted.

The OED gives a citation from 1551 using the form X'temmas (which makes it clear that we are dealing with an abbreviation). By 1660 we have Xtmasse and by 1721 the modern form Xmas had evolved.

But from the Etymonline website we can get a little further information. The Anglo-Saxon Chronicle of about 1100 uses the form Xres mæsse and apparently an in Old English an abbreviation for Christ was Xr, or interestingly Xp, which resembles the fist two letters Χρ of Χριστός. So if things had gone slightly differently we might have ended up with Xpmas and talked about ex-ar-mas cards.

What I haven't been able to find out so far is whether Xmas is peculiar to English or similar abbreviations are found in other languages.

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Richard Walker

"English As She Is Spoke"

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Edited by Richard Walker, Sunday, 15 Dec 2024, 00:17

I've often heard this expression, which I took just to mean a sort of proverbially bad phrase book. But to my surprise it turns out to be a real book.

Its true title was O novo guia da conversação em portuguez e inglez, "The new conversation guide in Portugues and English", published in 1855, but later dubbed 'English As She As Spoke" because of the startling phrases to be found in it. The section Idiotisms and Proverbs gives a flavour of the contents:

The necessity don't know the low. 

Few, few the bird make her nest.

He is not valuable to breat that he eat.

Its are some blu stories.

Nothing some money, nothing of Swiss.

He sin in trouble water.

A bad arrangement is better than aprocess.

He have a good beak.

In the country of blinds, the one eyed men are kings.

To build castles in Espagnish.

.. and so on. You can see this kind of stuff could get you some funny looks.

The general view is that the author had a Portuguese to French phrase book and a French to English dictionary, but did not speak English, so he picked a selection from the phrase book and then used the French to English dictionary to convert the French to English word by word.

                                    French phrase book                            French-English dictionary

Portuguese phrases   ---------------------------->     French      --------------------------------------->  English (sort of)

The original French phrase book was fine but the second stage, done in a very literal way by someone who did not know the target language, produced absurd and comical results.

Out of guilt perhaps, or possibly to lend an air of authority to his own work, the author of O Novo Guia gave the respected and learned author of the original French phrase book a credit on the title page. We don't know if the latter knew he had been listed as a coauthor but if he did he can't have been too happy about it.

English As She Is Spoke is easily found on the internet and there is more background information here at Wikipedia. There is also a good Rob Words video on YouTube.

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Richard Walker

Tongue-Twister

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This came up naturally in conversation and I found it quite hard to say. Try repeating it a few times

"My ex next door neighbour"

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Richard Walker

Knock-Knock!

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Richard Walker

My New Invention

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This is my new invention, which has many possibilities.

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Richard Walker

Teal (the bird not the colour)

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My brother photographed these teal a couple of days ago.


They are the smallest wildfowl species in the UK, weighing 300 or 400 g. Only 2,000 or so pairs breed here, but in winter nearly half a million birds migrate here from countries further north.

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Richard Walker

What I'm Reading

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Edited by Richard Walker, Friday, 6 Dec 2024, 23:47

More dipping into really. It's 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.  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.

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Richard Walker

Winter Rose Haiku

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Winter Rose


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






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