Knock-Knock!

Just saw this motivation video that began

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

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:

.. 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)

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.

Winter Rose

'Mistletoe' comes directly from Old English mistiltan. There's general agreement the -tan 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 was ἰξoς and this seems to be cognate withe the Latin viscum; originally it would have started with a 'w' sound ϝιξoς but the ϝ 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' and οἶκος 'house' (as in ecology and economy)

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

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. 

0    1    2    ...   955
🪣  🪣  🪣   ...     🪣 

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. 

Pyrex of the Caribbean

“Meet Antony. He used to be a cab driver”, said Tom exuberantly.

“Sydney and I have been chums for absolutely ages”, said Tom frenziedly.

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: 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.