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“Golden Bouquet”

Here's a rather neat bit of geometry. If we take a quadrilateral ABCD and join the midpoints of its sides, we get a parallelogram.

This theorem is named after Pierre Varignon (pictured, courtesy Wikipedia), a mathematician of the 17th and 18th century. 

Varignon was well-connected; it seems he knew Newton and Leibnitz, for example.

Now why should the theorem be true? Well suppose we concentrate just on EH and FG and draw in the diagonal BD, see below

Now there is a theorem that says if we join the midpoints of two side of a triangle the segment so obtained is parallel to the third side and half its length. Looking at triangles ABD and DBC tells us are half the length of the diagonal and parallel to it. Hence HEFG must be a parallelogram.

We can also see that a half is not special, for example if F, G, H and E had been one-third the way along the sides they lie on, instead of one half, we would still have obtained a parallelogram.

In my hedge grown these “pinkbells”, from bulbs I planted many years ago.

I just stumbled across this paradox which was only discovered (or invented?) quite recently. It's called the Brandenburger-Keisler Paradox and I'm still trying to get my head round it.

Next year will be a square year. 2025 = 45 x 45, and this is the only square year in this century, because 44 x 44 = 1936 and 46 x 46 = 2116.

By the way, here is a neat trick for squaring numbers that end in a 5. Suppose the number is X5, where X is some series of digits that form a number in their own right. Work out X times X+1, then pop 25 on the end.

Take 135 as an example. 13x14 = 182, so we get 18225, which is indeed 135 squared.

I slipped on a banana skin. It didn’t suit me so I took it off again.

What do you call a group of policeman standing in the middle of a field?

Copse.

You've probably met the Zen kōan of "clapping with one hand". Well, nalbinding is knitting with one needle. It is far older than knitting and examples from 6500 BCE have been found in a Judean cave. Wikipedia is full of information about the craft [1].

Nalbinding is also known from the Viking era, and the tradition is strong today in Scandinavia. Here's an attractive example fro the late 19^th century. [2]

The word means "needle binding", and the common root of "nal" and "needle" may perhaps be related to ancient Greek νήθειν = "spin". This connection is appealing but I couldn't find any strong evidence.

[1] https://en.wikipedia.org/wiki/N%C3%A5lebinding

[2] https://commons.wikimedia.org/wiki/File:1_par_vantar_n%C3%A5lbundna_av_vitt_ullgarn_med_dubbelvikt_krage_-_Nordiska_museet_-_NM.0005802B.jpg

I live in a very modest house. It doesn’t like talking about itself.

a woman who helps you cross a river?

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Bridget

Here’s a riddle I heard in the local Co-op.

What coat is best put on wet?