Suppose we have a long piece of wood with a uniform cross-section which is an irregular triangle (Fig. 1), so a straight cut at right angles to the length of the timber results in a triangle all of whose sides are different.

Show that it is always possible to make an oblique cut at an angle in such a way that the section obtained is an equilateral triangle (Fig. 2).

This comes from the Turkish Maths Olympiad 2000 and I read about it here, p. 141. 

An outline proof appears in the comments.

This plant, which growing profusely in my street, is Red Valerian, Valeriana Rubra (or Centranthus ruber). Originally from the Mediterranean, it has been introduced into many other parts of the globe and become widely naturalised.

The genus Valeriana was named by Linnaeus, after the Roman Emperor Valerian. Valerian means "Worthy or "Strong" and descends from the same root as "Value", "Valid", Valour" etc. Linnaeus chose this name because another Valerian species, Valeriana officinalis has been used since the time of Hippocrates as a herbal remedy with a range of uses, so I suppose he felt it deserved the description of "Worthy".

Two interesting facts about V. officinalis.

1. It's attractive to cats, like catnip.

2. An old Swedish country superstition was that (it is so fascinating I shall quote it in full)

'… a bridegroom stands in dread of the envy of the Elves, to counteract which it has long been a custom to lay in the clothes on the wedding day certain strong-smelling plants, as garlic or valerian. Near gates and in crossways there is supposed to be the greatest danger. If any one asks a bridegroom the reason of these precautions, he will answer : " On account of envy." And there is no one so miserable whose bride will not think herself envied on her wedding day, if by no others, at least by the Elves.'

The elves would spirit the groom away

' The bride sits ready in her bridal bower, in anxious ex pectation and surrounded by her bridesmaids. The bride groom saddles his grey steed, and clad in knightly attire, with his hawk perched proudly on his shoulder, he rides forth from his mother s hall, to fetch home his bride. But in the wood where he is wont to hunt with hawk and hound, an elfin maiden has noticed the comely youth, and is now on the watch for an opportunity, though for ever so short a time, to clasp him to her breast in the flowery grove; or, at least, to the sweet tones of their stringed instruments, lightly to float along with him, hand in hand, on the verdant field. As he draws near to the elf-mount, or is about to ride through the gateway of the castle, his ears are ravished with most wondrous music, and from among the fairest maidens that he there sees dancing in a ring, the Elf-king s daughter herself steps forth fairer than them all, as it is said in the lay :

The damsel held forth her snow-white hand : " Come join in the merry dance with me."

If the knight allows himself to be charmed, and touches the fascinating hand, he is conducted to Elfland, where in halls indescribably beautiful, and gardens such as he had never beheld, he wanders about, on his Elf-bride s arm, amid lilies and roses. If at length the remembrance of his mourning betrothed enters his mind, and the Elves, who do not deliberately desire evil to mankind, are moved'

From 

Northern mythology : comprising the principal popular traditions and superstitions of Scandinavia, North Germany, and The Netherlands, volume 2  

SCANDINAVIAN POPULAR TRADITIONS AND SUPERSTITIONS (1851), VOL: 2, OF ELVES, p. 62 https://www.artandpopularculture.com/Northern_mythology_:_comprising_the_principal_popular_traditions_and_superstitions_of_Scandinavia,_North_Germany,_and_The_Netherlands,_volume_2

For some reason I wondered about the origins of "go" and when I looked it up found a discussion of why the past tense of "go" is "went" rather than "goed". It is rather strange.

In Old English "go" was gan, the ancestor of the modern word, and it already had an irregular plural eode, a word of unknown origin.

At that time "went" was the past of wendan, connected with modern wind (as in turn). "Wend" has survived into Modern English as "wend", as in wend ones way. Nowadays the word has narrowed its meaning to imply travelling in a circuitous and roundabout way and has a rather archaic ring, but originally seems to have meant wind in the general sense.

"Wend" had a past tense "went" (like spend -> spent; send -> sent; lend -> lent; rend (meaning rip) -> rent and probably others) and for some reason is Early Middle English this started to be used in place of eode, which was eventually replaced, and disappeared from the language.

But why was "goed" not adopted? One possibility could be to avoid confusion with "good" which would at that point have had a vowel sound like that in "oak" and a 'g' like that in "go". Just a theory, I don't really have a grasp on these sound changes.

As a wordplay enthusiast I've often mused over what words can be formed by combining the symbols of the chemical elements.

For example, Actinium-Actinium-Iodine-Arsenic would give "acacias" and Barium-Oxygen-Barium-Boron-Sulphur would give "Baobabs".

A while ago I did try to come up with some  examples by hand but only found a few, so I decided to write a little program in Python.(see below[*]). For simplicity I just used lowercase letters e.g. "h", "he", "li" etc. but putting back the capitals would not be too hard. I also excluded the artificial transuranic elements (26 found to date) but again adding them would be easy enough. My reason for omitting them is that their name and symbols are a bit less familiar.

Also note it uses a "greedy" algorithm and if the first two letters is a valid symbol it chooses that first, so it can't handle "those", because it finds "th" (Thorium), then "os" (Osmium), and now the "e' is orphaned. To overcome this I'd need to add some backtracking capability so the program gets a bit more complex.

I analysed an open-source word list of 113,810 words and found 13,435 - let's call them "elementary" - words, about 12%. We'd expect the chances of a word being elementary to fall off as words gets longer, because only 92 one- or two-letter combinations of letters are symbols of chemical elements but the total number of possible one- or two-letter combination is 702, so only about 1 in 8 correspond to elements.

Nonetheless I found some surprisingly examples, with the champion being the 16-letter "counteraccusations". Here are some other long ones.

acacias
acarpous
accepters
accessions
accountancy
accurateness
accuratenesses
articulatenesses
counteraccusation

"counteraccusations" represents Cobalt Uranium Nitrogen Tellurium Radium Carbon Copper (Cu) Samarium Titanium Oxygen Nitrogen Sulphur; 11 distinct elements which I imagine is a record.

What about whole sentences? It's not easy to make up natural-sounding ones, but here is my attempt at one from a scientific setting. I've added proper punctuation to make it more realistic.

"Pop both new boxes in lab one Gabby, ta."

[*] Program follows: list of symbols, then function to test words  then sample function calls.

elements = [
    "h", "he", "li", "be", "b", "c", "n", "o", "f", "ne",
    "na", "mg", "al", "si", "p", "s", "cl", "ar", "k", "ca",
    "sc", "ti", "v", "cr", "mn", "fe", "co", "ni", "cu", "zn",
    "ga", "ge", "as", "se", "br", "kr", "rb", "sr", "y", "zr",
    "nb", "mo", "tc", "ru", "rh", "pd", "ag", "cd", "in", "sn",
    "sb", "te", "i", "xe", "cs", "ba", "la", "ce", "pr", "nd",
    "pm", "sm", "eu", "gd", "tb", "dy", "ho", "er", "tm", "yb",
    "lu", "hf", "ta", "w", "re", "os", "ir", "pt", "au", "hg",
    "tl", "pb", "bi", "po", "at", "rn", "fr", "ra", "ac", "th",
    "pa", "u"
]

def parse(word):

    result = []

    while len(word) > 0:

        token1 = word[0:1]
        token2 = word[0:2]

        # Look for two-letter symbol first
        if token2 in elements:
            result.append(token2)
            word = word[2:]

        # If not check for 1--letter symbol 
        elif token1 in elements:
            result.append(token1)
            word = word[1:]

        else:
            return 'fail'
        
    return result

print(parse('cream'))
print(parse('scone'))

Equilateral triangles ABC and CDE share a common vertex at C. Prove that AD and BE are equal in length.

Solution tomorrow.

Every now and again someone comes up to me in a pub or a hotel or a classroom and sketches this little picture.

"Can you draw this", they say, "without going over the same bit twice or lifting the pen off the paper?"

"I can", I say, and then I try to explain how I can be so confident without even picking up the pen. Here's how it goes.

First identify each point where two of more lines meet.

Now count how many lines meet at each point.

For it to be possible to trace the figure it's necessary that either:

• All the numbers are even, or;
• Two and only are odd (as is the case here) and you start from one of these and end at the other. 

This is because every time you pass through a point you add 2 to the count. If you start and finish at the same point then an extra 1 is added when start and another 1 again when you finish, so all the counts are even. Otherwise if we start and finish at different points, both get an extra 1, so their final count will be odd but all the other counts even.

Providing one of these holds it can be proved (but I'm not going do so here, it's not that hard but a bit too long) that the figure can be traced. And here's a solution (not the only one). The points with an odd count are highlighted.

These ideas were first articulated by Leonhard Euler in 1736. A problem that must have been going the rounds at the time concerned the seven bridges of Königsberg (in Prussia then; now Kaliningrad in Russia). The puzzle was: is it possible to find a tour that crossed each bridge exactly once.

Here are the bridges as shown in Euler's original paper, which you can find in translation here.

Out of interest I looked for the bridges on Google Earth and as you can see only five are still there; b and d were presumably destroyed in WW II. I hope I have identified the other five correctly.

Now if we take A, B , C and D to be points, and the original seven bridges a - g as lines connecting the points, we get this figure, and the problem of the Königsberg bridges becomes one of tracing the figure without crossing any bridge twice or taking the pen off the paper.

All four of A - D have an odd number of lines meeting at them, so from the discussion given earlier the puzzle of the seven bridges has no solution, although with only the five bridges that have survived it does become possible to make a tour of the desired kind, starting at A and ending at D, or vice versa.

Euler called his paper, which has since become very famous, "SOLUTION OF A PROBLEM IN THE GEOMETRY OF POSITION" and it gave rise to the branch of mathematics now called graph theory: nothing to do with x and y-axes, or plotting lines and parabolas etc., but concerned with the properties of arrangements consisting of points (vertices) connected by lines (edges). As well as being of great intrinsic interest and beauty, graph theory is a vital tool in the design of the algorithms and data structures behind the computer programs that play such an important role in modern life.

My friend heard a song by Duffy and when she sang

what he heard was

"Mercy" makes better sense, people don't often beg for birdseed, but that's not necessarily how Mondegreens work.  In The Language Instinct Steve Pinker mentions Mondegreens and notes that the mishearing is often fits the context less well than the original word and this seems be an example of that.

Yet perhaps a singer begging for birdseed does have some kind of underlying logic, when you think about it.

"AI Overview" in Chrome says a polyhedron is:

"a three-dimensional geometric shape with flat polygonal faces, straight edges, and sharp corners or vertices"

which is not a bad definition in my view. Some of these polyhedra are familiar to us; for example here is a back-of-an-envelope sketch of some with 4, 5 and 6 faces.

Note that we are not concerned here with the particular angles or side lengths, just with the topology: how many faces and vertices there are, how many sides each face has, and what faces fit with what other faces around each vertex. 

The solid with 4 faces - a tetrahedron aka triangular pyramid - is the only possibility for 4 faces, and for 5 faces there are exactly two polyhedra - the pyramid with a quadrilateral base and the triangular prism.

The two hexahedra shown are not the only ones however. There are 7 altogether; can you find some or all of the remaining ones? Solution in the Comments.

As the number of faces grows the number of possible polyhedra climbs exponentially [1].

Numbers up 10 faces are exact, but those from 11 on are only estimated.

Amazingly Steven Dutch has enumerated and classified all 2606 enneahedra, finishing on 2 November 2016 [2].

Here's a quick question to finish - what is a regular hexahedron better known as?

References

[1] https://oeis.org/search?q=A000944&language=english&go=Search

[2] https://stevedutch.net/symmetry/polynum0.htm

Malfatti asked how to find three circles in a given triangle, such that each circle would be tangent to the other two cirrcles, and to two sides of the triangle. It seems to have been thought at one time that this would also be the solution to the (different) problem of packing three (not necessarily equal) circles into a triangle so as maximise the are covered by the circles.

For an equilateral triangle Malfatti's original problem is solved by the arrangement in Figure 1.

Intuitively it seems quite plausible that might at the same time maximise the area covered. There doesn't seem to be any way we can make any of the circles bigger without the others becoming smaller, and we might expect the case when the circles are equal to be optimal.

But it isn't; the arrangement of Figure 2 covers a bigger area.

Rather pleasingly the small circles are one-third the size of the big one, as illustrated by te three dotted circles.

The difference is not great, but after some trigonometry and algebra it emerges that Figure 2 is the clear winner, covering 73.9% as opposed to Figure 1 with only 72.9%. 

This set me thinking: what if we replace the equilateral triangle with a square and asked for four circles that cover as much of the square as possible?

The arrangement analogous to Figure 2 appears in Figure 3.

After a bit of calculation, we find this covers 85.5%, and based on our earlier experiences we'd expect this does better than using four equal triangles.

And indeed it does, and we need no calculations; here is a "look and see" proof.

This just amounts to four half-scale copies of Figure 3, but minus the three small circles. So in each smaller square we are covering less than the 85.5% of Figure 3, and the overall percentage covered must also fall short of this value.

This is not a proof that Figure 3 is the best possible of course; only that it is better than Figure 4. Maybe some other arrangement might achieve more than 85.5%.

My brother snapped this unusual cloud formation.

The Cloud Appreciation Society classified it as "Altocumulus stratiformis radiatus, often seen as a precursor to changing weather patterns" and sure enough the weather did change.

I watched a video on YouTube about a bee expert relocating a swarm of bees. After he'd skilfully coaxed them into a special cardboard box, he was asked how many he thought there were, and he said, by the weight of the box, about 20,000.

This reminded me of an old story about a man who had to transport a large number of canaries on perches in the back of a lorry. But the combined weight of the canaries was more than the lorry's axles could safely bear.

To get round this, he got another man to travel in the back with the canaries and stir them up with a pole at frequent intervals, on the principle that as long as they were flying around they wouldn't put any weight on the lorry's axles.

Would this scheme work?

This charming little tree with its delicate foliage is a silver wattle, Acacia dealbata, and a newcomer in our garden.

A. dealbata is a fast growing evergreen, native to south-east Australia, and can reach 30 m in height, so this little tree will need watching.

The flowers of A. dealbata are a mass of bright yellow, picture in the photo from Wikipedia.

A relative, the golden wattle Acacia pycnantha, also has bright yellow flowers, and is the national flower of Australia. I've read a few stories set in the outback and wattles often got a mention but until now was pretty hazy about what exactly there were.

Acacias belong to the pea family of plants and there about 1,000 different species in Austraila. and Africa. The name acacia and the common name wattle are both interesting words etymologically speaking. Acacia is from the Ancient Greek name for the plant, ἀκακία, "akakia", via Latin. The origins of the Greek name are unclear however; it might be related to PIE *ak-, "sharp", "topmost", as found in words like acute, acid, acropolis, acrobat, acronym.  But other theories suggest it was borrowed into Greek from a pre-Greel language spoken in the region.

Wattle seems to derive from the wattle in the phrase "wattle and daub", a traditional way of constructing walls with a framework of branches roughly plastered with lime plaster or mud, perhaps with horsehair in it. The trees were called wattles in Australia because there they often provided the required branches. Wattle itself comes from a OE word watel, "hurdle", and this may have a connection with weave, but there are different theories and the ultimate origin is unclear.

Image credit: Eugene Zelenko, see here