If Puerto Rico were to become the 51st state—and granted, that’s at least four ifs away—federal law requires that a new star be added to the American flag. One can’t help but wonder: Where would we put it?
There’s a flag generator which allows you to vary the number of stars from 1 to 100. There is no “valid pattern” for 29, 69 or 87 stars — none of the desired symmetries are possible — and a few of the other patterns look like “why don’t you admit two states at a time” (like 79, 89 and 92)
How to multiply, Roman-numeral style. A Different Kind of Multiplication
It is often said that an important advantage of the decimal notation over the Roman one is that makes multiplication of numbers much easier. Adding CLXXVII to XXIII may be relatively straightforward–but how about multiplying the two?
The Virtuosi: My Pepsi* Challenge
The basement of the Physics building has a Pepsi machine. Over the course of two semesters Alemi and I have deposited roughly the equivalent of the GDP of, say, Monaco to this very same Pepsi machine (see left, with most of Landau and Lifshitz to scale). It just so happens that Pepsi is now having a contest, called “Caps for Caps,” in which it is possible to win a baseball hat. There are several nice things about this contest. Firstly, I drink a lot of soda. Secondly, I like baseball hats. So far so good. Lastly (and most important for this post), is that it is fairly straightforward to calculate the statistics of winning (or at least simulate them).
On the surface this is a lesson in basic statistics and simulations. But the other lesson here is that physics graduate students have been conditioned to not value their time paricularly highly.
Built on Facts: Sunday Function
Take an integer – one of at least several digits – and multiply it by itself twenty times. The result is going to be some really gargantuan number. Take the last 10 digits of that number. That’s the output of our function, which we’ll call h(n).
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The resistance can use this property of hash functions to make their resistance network more secure. Instead of distributing a list of all the agent’s passwords, the resistance can distribute a list of the hashes of their passwords. Thus if Bob knows that Alice’s hash is 7001140801, Alice can verify her identity by saying that her password is 314159, which has that as its hash. But if a Nazi double agent (let’s call her Eve) has managed to steal the list of hashes, she still can’t impersonate Alice. Eve doesn’t know what password to use to generate that hash. She could try thousands or millions of guesses and hope that eventually she found one with a hash that matches Alice’s hash, but with all the possible hashes that would be a herculean task.
Non-Normalizable Probability Measures for Fun and Profit
An eccentric benefactor holds two envelopes, and explains to you that they each contain money; one has two times as much cash as the other one. You are encouraged to open one, and you find $4,000 inside. Now your benefactor — who is a bit eccentric, remember — offers you a deal: you can either keep the $4,000, or you can trade for the other envelope. Which do you choose?
[T]he Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).
Now suppose infinitely many new guests arrive, sweaty and short-tempered. No problem. The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on. This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.
It’s the end of the From Fish to Infinity series to which I had previously linked.
Take the numbers 1 2 3 4 5 6 7 8 9 and make them equal 100 by placing any mathematical symbols you like between them.
Saw this in an infographic (I was just looking, really).
Wait, what? The number of men + women arrested doesn’t add up to 100%? Is there a third gender, called “customers?” Or, all customers are hermaphrodites?
Mathematics Reveal Universal Properties Of Old Rope
Despite rope’s obvious geometric properties, the art of rope making has been strangely neglected by mathematicians over the centuries. Today, Jakob Bohr and Kasper Olsen at the Technical University of Denmark put that right by proving the remarkable property that ropes cannot have more than a certain number of turns per unit length, a number which depends on the diameter of the component strands.
And that’s just the start. They go on to show that a rope with a smaller number turns than this maximum will always twist in one direction or another under tension.