I sent my daughter this
What do cats do when they change their genes?
Mew-tate!
Her friend saw my pun and raised me
You might think this is funny, but cat puns really freak meowt.
I sent my daughter this
What do cats do when they change their genes?
Mew-tate!
Her friend saw my pun and raised me
You might think this is funny, but cat puns really freak meowt.
I'm a big doodler. Last night I drew some dots in a rough octagon and started drawing lines joining pairs of dots.
Then I wondered how many lines I could draw before I formed a triangle. I don't mean a triangle formed by three lines (although that's an interesting question too), but one with dots at its three corners.
With any problem where you have a series it's always a good idea to look at the small cases. Sometimes it can be misleading though. Still I drew some little sketches
It's easy to be sure the count is right for n = 2, 3 and 4, but even by 5 it gets less obvious. Is there a general formula? I wondered if this was a novel puzzle (and hoped it was); sadly it turns out no, it was investigated by Mantel in 1907. There are various proofs of the result Mantel found, but they are more or less technical and not intuitive. So I wondered if I could present the basic idea of one proof in a more accessible way.
With 5 dots consider a pair x and y that are joined by a line. That leaves 3 others and if we think about any one of these, let's call it z, we see x and y cannot both be joined to z. Otherwise we would have a triangle! So the maximum of lines from either x and y to the other dots is 3. Finally focusing on the 3 other dots, the maxiumum number of lines possible amongst these is 2, otherwise we would have a triangle.
Adding this all up as shown the greatest possible number of lines is 1 + 3 + 2 = 6.
Let's see if we can generalise this idea. Suppose we have n dots and the maximum number of lines that can be drawn without forming a triangle is f(n).
Following the same argument as before, if we are to avoid a triangle, we can have one line joining x to y, a maximum of n - 2 lines joining either x or y to one of the remaining n - 2 dots, and f(n - 2) lines amongst tos n - 2 others. So
f(n) = 1 + n - 2 + f(n - 2) = n - 1 + f(n - 2)
So we can immediate calculate further values
f(6) = 6 - 1 + f(6 - 2) = 5 + f(4) = 5 + 4 = 9
f(7) = 7 - 1 + f(7 - 2) = 6 + f(5) = 6 + 6 = 12
f(8) = 8 - 1 + f(8 - 2) = 7 + f(6) = 7 + 9 = 16
So now we have (for completeness we include n = 1 with 0 lines)
0, 1, 2, 4, 6, 9, 12, 16
Can you spot a pattern?
This other-wordly organism is a lichen, "Pixie Cups", photographed by my brother Simon at Sandy Lodge in Bedfordshire.
Thius seems to be related to the so-called "Reindeer moss", also a lichen of the genus Cladonia, which contains 276 species. Some are quite startling in appearance, see here.
I’ve got a new job; vaccinating people. Can’t wait to get stuck in.
Winter jasmine
Here already
What a short year
Repeat five times after me
"On an anonymous ominous omnibus."
I like this Mondegreen. Although it's entered into the folklore, and is found on numerous websites, it has the ring of a genuine Mondergreen; a bit like "Gladly, the cross-eyed bear".
"We three kings of porridge and tar."
I worked with a men’s neckwear company for many years. But it’s all over now, and we've severed our ties.
I've just (10 mins ago) bought a marvellous book
THE LANGUAGE LOVER’S
PUZZLE BOOK Lexical Complexities and Cracking Conundrums from Across the Globe ALEX BELLOS
Bellos, Alex; Bellos, Alex. The Language Lover’s Puzzle Book: Lexical perplexities and cracking conundrums from across the globe.
It's only just been published but what sold it to me was Alex's talk from the Royal Institution
https://www.rigb.org/whats-on/events-2020/november/public-the-puzzle-of-language
It also made me subscribe to the Royal Institution. I remember the physical building from visits in my teenage years, where we sat in the same room where Michael Faraday lectured, and the lecturer stood behind the same laboratory bench. I've watched a few lectures on YouTube but I've probably missed some good ones, so now I'll get alerts.
The police caught me in their dragnet. First they threw an edible water plant over me. Then they shouted, “You’re under a cress!”
Why are the other 25 Ectors being ignored? We demand the truth.
Suppose inhabitants of Diss in Norfolk UK had been left without voting rights, as a result of some historical mix-up, and this anomaly was corrected. Then they’d be Diss-enfranchised. Hmm.
Davies
What is this life if, full of care,
We have no time to stand and stare?
Me
I often have the time to sit and muse,
Especially when I’ve had a lot of booze.
There actually is a place called Epsom Downs. Think about it.
We hear a lot about soldier ants. But what about noncombatants?
The famous American inventor Thomas Edison said genius is 1% inspiration and 99% persperation. He never gave up.
For example he invented the heavy bulb, a total failure.
So he moved on to invent the light bulb, which did a lot better.
Died at sea from backing hidden word? (7)
Responsive device when a person is out (11)
See comments for solution.
Please rate this porridge, on a scale of 1-5 stirs.
🥄🥄🥄🥄🥄
What Came Before Toucans?
Onecans!
I couldn't think of a word meaning the same as kayak. Canoe?
My policy about sweeping away cobwebs is put it off until the last possible moment. I call it Dust in Time.
It’s hard for singers of classical music to sing so listeners can pick up the words.
They are usually expected to sing in several languages not their first, and even native speakers of the language the performance is in often struggle to understand the singer.
Following on from the Berkeley research described a post or so back, maybe subtitles ahead of each phrase would help people pick out the words.
This blog might contain posts that are only visible to logged-in users, or where only logged-in users can comment. If you have an account on the system, please log in for full access.