There is an elegant proof of that theorem that uses the pigeon-hole principle, and yet I find the result ... charming.
It can also be generalised to other bases, etc. There's more going on.
But this question is not about this. This question is about other examples of the PHP giving a nice, clean, elegant proof of a non-obvious statement.
Any one have some good examples of the pigeon-hole principle being used to prove something that seems surprising?
For example:
Theorem: For every positive integer N, there is a multiple kN such that the decimal representation of kN consists of only 0s and 1s.
Examples:
13 -> 1001
There's a paradox wherein building a new route can increase congestion ... but my search fu is failing me.
Help?
This is a hill I'm not willing to die on, but it is something I wonder about.
People talk about "solving" rubik's cube, when really they are talking about "restoring" the cube.
I would think that "solving" implied inventing the algorithms.
* Solve => invent algorithm
* Restore => Follow an algorithm
But this battle is lost, and people will always talk about "solving the cube".
So ... what pithy terms could/should be used for the process of inventing algorithms for restoring the cube?
Some weeks ago I received an email from the Manchester Airport Group saying that their systems had been breached and hackers had obtained email addresses and other details.
Yup ... today started to get spam addressed to the unique email I'd created to register with their wifi.
I always create a new, unique email address for every service that requires one. Not everyone can do the same, but it's worth trying to work out how to do it.
In today's mysterious adventures ...
I have a minimalist document in HTML. It's a table, renders perfectly in the Firefox and the DDG browsers.
Some cells have bgcolor set, and that shows up nicely.
I want to export to PDF and the immediately obvious thing to do is just hit Ctrl-P in the browser and select PDF output.
But the preview shows it to be in B&W.
Any obvious reason for that?
Do I know anyone in the Bristol area?
An author friend of @virtuosew is giving a talk on Thursday October 1st.
It's not for everyone, but if you can have a look to see if it's for you, and possibly boost to get outside my/our bubble, that would be much appreciated.
https://www.eventbrite.co.uk/e/author-talk-alison-baxter-at-bishopston-library-tickets-1997137169034
Every time you clean something you just make something else dirty.
Hmm. I should probably make an account and verify that I've got the right answer ...
On the off-hand mention of a friend, I got Nerd-Snup by Euler Project 132.
Fun diversion.
"People think mathematics is complicated. Mathematics is the simple bit. It's the stuff we can understand. It's cats that are complicated." -- John Horton Conway
"You know, the very powerful and the very stupid have one thing in common. They don't alter their views to fit the facts. They alter the facts to fit their views." -- Doctor Who (1977)
"You don't say, 'I've done it!' You come, with a kind of horrible desperation, to realise that this will do." -- Anthony Burgess, on finishing books
RE: https://mathstodon.xyz/@johncarlosbaez/117185013638534442
Not sure I've ever had a dream like this ...
"A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools." -- Douglas Adams, Mostly Harmless
"English is weird: 'I'm sorry' and 'I apologise' mean pretty much the same thing. Except when you're at a funeral." -- Demitri Martin
"A mathematician is a device for turning coffee into theorems." -- Renyi (although often (incorrectly) attributed to Erdos)
Write down 5 different digits in a row such that each consecutive pair is a square number.
28423 fails because:
* 2 is repeated;
* 28 is not a square;
* 84 is not a square;
* 42 is not a square;
* 23 is not a square.
*Please* avoid spoiling the fun for others by putting your answers in a DM, or at the very least behind a CW.
Thank you ... and enjoy !!
Nice puzzle:
-> ...
(Seen in Chris Smith's Maths Newsletter)
Peter Cushing’s Toy Soldiers – No Rerolls:
https://norerolls.co.uk/2017/12/03/peter-cushings-toy-soldiers/