Personal Blogs

There is a class of geometric problems that ask about covering shapes with other shapes. Some have practical applications but mainly they are studied for interest, and because they can be easy to state but unexpectedly hard to solve.

One type of problems concerns covering a circle with 1 x 1 squares. If we use 1, 2, 3... squares, what is the biggest circle we can completely cover in each case?

Here are the optiomal arrangements for 1, 2, and 3 squares

I constructed the first two in GeoGebra but the third is a sketch based on a diagram by Erich Friedman.

These have been proved to be optimal. Here are even sketchier pictures of the other two cases where the best possible arrangement is known with certainty.

You may see a pattern here, something like "when the number of square is a square number 1² =1, 2² = 4, 3² = 9 etc, the best arrangement is just a square grid. Obvious really.

But it's not true! When we get to 16 the pattern breaks down and the surprising arrangement below turns out better. I didn't say best, because this is just the best known, and it might be possible to improve on it. By this point I'd so many diagrams I decided to cheat and simply copy Eric's amazing diagram.

Kind of crazy but it does have a line of symmetry. Erich Friedman's github page here gives best known solutions up to 18 squares and they are all symmetrical about a line, but this not a given; there may be improved solutions with now symmetry, it's perfectly possible.

Here's a puzzle to end with; if we go back to the case of two squares, whar is the radius of the circle? You can look it up on Erich's page of course, but can you work it out?

Propositiones ad acuendos juvenes, "Problems to sharpen the young", is an early collection of recreational maths problems, generally attributed to Alcuin of York (roughly 735 – 804), although the authorship is not certain.

In it we find some familiar problems, such as the one about the man who has to transport a wolf, a goat, and a cabbage across a river. The wolf can't be left alone with the goat nor the goat with the cabbage.

"De lupo et capra etfasciculo cauli - A wolf, a goat and a bunch cabbages. A man had to take a wolf, a goat and a bunch of cabbages across a The only boat he could find could only take two of them at a time. he had been ordered to transfer all of these to the other side in condition. How could this be done."

After more than 1,000 years this puzzle remains seems to have retained its novelty and Googling "wolf, goat, cabbage" gets over a million hits.

Here's another I liked

De duobus hominibus boves ducentibus - Two men leading oxen. Two men were leading oxen along a road, and one said to the other: "Give me two oxen and I'll have as many as you have". Then the other said: "Now you give me two oxen and I'll have double the number you have." How many oxen were there, and how many did each have?"

The translations I'm reading are from Hadley and Singmaster [1] and a recent book by Marcel Danesi [2].

[1] Hadley, J. and Singmaster, D.  Problems to Sharpen the Young. The Mathematical Gazette , Mar., 1992, Vol. 76, No. 475.

[2] Danesi, Marcel. (2024) Alcuin's Recreational Mathematics: River Crossings and other Timeless Puzzles.

I've always been fond of hot food - hot in the sense that chillies are hot -  so I thought I'd list all the hot spices and vegetables I could think of, and see what was in them that made them hot. I was expecting some degree of overlap, but it was more than I had thought.

Essentially there are just two sort of "hot" food substance. The first shares a family of active ingredients that are chemically related

All the other "hot' things I thought of (Horseradish, Mustard, Radish, Wasabi) belong to the cabbage family and what makes them hot is Allyl isothiocyanate (aka mustard oil). Here's a model of the molecule, from Wikipedia:

Grey atoms are Oxygen Hydrogen, the black Carbon the blue Nitrogen, and the yellow Sulphur.

I found this peach-leaved bellflower (Campanula persicifolia) at the side of the road. It's a British wild flower but the species is also grown in gardens, so this may have been an escape.

Looking the plant up, I found its blue colouration and the blue and violet colour found in all 550+ known Campanula species is due to a compound called Violdelphin, one of a group of compounds, called flavonoids, which plants manufacture, and which benefit plants in a variety of ways. 

In the UK six or more other bellflowers grow wild, including harebells and clustered bellflowers.

I had rollmops for lunch today and decided to look into the origin of the name. A rollmop as you probably know is a pickled herring fillet, rolled up and usually held together with a short wooden skewer. I got the roll bit but the mops was a mystery.

I looked it up in the OED and it turns out it's a 19c borrowing from German Rollmops, which is Roll + mops, meaning 'pug', so Rollmops literally means 'roll pug'. The plural is Rollmöpse, Rollmops being the singular, and English rollmop a back formation.

This is a bit like 'pea', which is a back formation from 'pease', as in pease-pudding. 'Pease' sounded like a plural and so people assumed one of them would be a pea.

Mr Paradock: A comptometer then. What does it matter? Tell me to do something. Go on. Feed me some data.

From the play One Way Pendulum by N F Simpson (1959)

You will know that the earliest Computers were not machines but people. For a while I was one of those human computers. Working for the London Brick Company I was paid 16s an hour, good money in those days. 

What did I do? Computed of course. There were scores of us in a big room, performing various tasks, but mine was drawing up a weekly report on brick sales. It went something like this.

People brought me records of how many bricks of different varieties (such as Flettons or Wirecuts) had been supplied from a particular factory (such as Stewartby). I merged all these into a big spreadsheet and then worked out the totals. Manually, note; we weren't given mechanical adding machines and pocket calculators didn't exist yet.

We used different coloured inks to distinguish different sorts of brick and pens with very small nibs so we could write very small figures.

Millions of brick were involved but the numbers had to balance, to the exact brick, and to check our working we divided all the numbers in a row or column by 13 using mental arithmetic, added up all the remainders, and checked that the sum agreed with the remainder on dividing the row or column total by 13.

Already it was obvious that the skills this work required would soon be obsolete. In the middle of the office was a small row of Comptometers, like that below, that only specially trained staff could use.

They were the latest thing but then in the twinkling of a silicon chip they too were out of date.

• N F Simpson belonged the movement known as the Theatre of the Absurd. The play concerns the eccentric Mr Groomkirby who wants to build a replica of the Old Bailey in his living room. The Comptometer is introduced in Act II.

In case you don't know, a mondegreen is when a listener puts the wrong words to what they hear. The term mondegreen was coined in 1954 by Sylvia Wright, whose mother used to read her an old ballad:

Ye Highlands and ye Lowlands,
Oh, where hae ye been?
They hae slain the Earl Amurray,
And Lady Mondegreen.

Who was this mysterious Lady Mondegreen? Sylvia Wright had a vivid mental picture of her and her sad end.

But the was no such person; the last line is actually:

And laid him on the green.

And from Wright's article the word mondegreen made it into our language and can now be found in the OED.

I've posted about modegreens before, they fascinate me , and this morning I came across a nice example. Someone said:

"I think I'm overdosing on drinking tea." 

Makes sense I thought, we all know too much caffeine can be bad for you.

But what they'd actually said was:

"I think I'm overdosing on Vitamin C." 

But they weren't, it's OK.

A square of side length 3 units has a vertex at the centre of a larger square whose side length is 4 units. Find the area of the shaded region.

In the bird reserve the evening before last I heard a very striking song, which the BirdNET app reported as "almost certainly" Cetti's warbler. Comparing the song with online recordings confirmed the identification.

Cetti's warblers don't seem to have bred in the UK until the 1970s, when they first appeared in Kent. Since then, possibly as the result of global warming, they have pushed as far as Wales, the north of England, and recently Scotland. There are now perhaps 5,000 or more breeding pairs across the UK.

I didn't see the bird.They are hard to spot; small and inconspicuous and in the words of the RSPB like to skulk in patches of scrub (although "skulk" seems a bit pejorative to my mind).

Traditionally people talked about bird watching but when it comes to birds like warblers bird listening is more appropriate, because warblers recognise potential mates not by conspicuous plumage, but by conspicuous song. I'm not very good at recognising birdsong, but the phone app is, and it's opened up a whole new world for me, where I can observe birs even though I can't see them.

And who was Cetti? Francesco Cetti was a Jesuit, mathematician, scientist and naturalist who published three volumes on the natural history of Sardinia in the 1770s. This is the volume on birds,

The bird was collected on Sardinia and described by Alberto della Marmora, who named it Cettia cetti in Cetti's honour.

Fun Fact: The eggs of Cetti's warbler are bright red.

Another unusual word I owe to the "Word Watching" item in Times 2.