Hey, long time without posting. Hope this will change drastically in the near future. In the meantime… can anybody tell me what’s that!? :)
Category Archives: puzzles
Let me count the ways…
Alice was so bored, waiting for a message from Bob, that she started to play with the five white rabbits she had got from the Queen of Hearts. She tried to figure out in how many ways she could split her rabbits in groups, like 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so, for 5 rabbits, 7 ways.
She decided to count the partitions in which no group contained more than 3 rabbits… in our example, there are 5. And then, she counted the partitions with no more than 3 groups. Amazingly, although the groups were not the same, the two numbers coincided, also 5.
Alice wondered… She wonders all the time (why?). Is that a coincidence?
A few unrelated questions around π…
- Why is it true that π=80?
- Why on Earth did we define π as we did, instead of giving a nice symbol to 2π? Life would be much easier… So many less factors 2 in our books… A quadrant would be just π/4, not the nonsensical π/2… Can you see any notational advantage? Read this for more info.
- Do you recognize this sequence: 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, …?
- Why should I have posted this yesterday?
OK, let’s keep it short. And thanks to S.N. Santalla…
Update (March 17) My birthday date appears at position 45,260,128 of π, not counting the initial 3. When was I born? ;) Hint. (Via Pepe Aranda) Moreover: possession of all digits of π makes you infringe all known copyright laws… Do you know why?
The old man stared at us and spoke thus: “Those were the days, in Mod Island… We lived surrounded by beautiful colorful chameleons… I remember that, the day when I reached, 17 of them were blue, 15 were red and 13 were yellow. But, whenever two chameleons of different colour met, they would start a funny dance after which both would change to the colour which neither of them had. For example, if a blue and a red chameleon met, after the dance they both would become yellow. These dances took place for a long time, until all the chameleons became the same colour, and we knew it was time to leave…”.
“You’re lying”, said Alice. The old man grinned… “How are you so sure?” And Alice replied back, “Because, with your numbers, it’s impossible that all chameleons become the same colour”.
Is Alice right?
More about Euler’s crazy sums
Recently we talked about how Euler managed, through some magic tricks, to find that the sum of the inverse of the square numbers is … That was a piece of virtuosismo, but even geniuses sometimes slip up. This one is funny. You may know the sum of a geometrical series:
(There are many ways to understand that formula, I’ll give you a nice one soon). Now, using the same formula, sum the inverse powers:
So far, so good. Now, imagine that we want the sum of all powers, positive and negative:
Amazing!!! The sum of all powers is zero!! Nice, ein? Euler concluded that this proved the possibility of the creation of the world from nothing, ex nihilo. So, this is easy… what is the problem with this proof?
BTW, I read this story and many others in William Dunham’s book The master of us all…
Time travel from classical to quantum mechanics
I would like to return to the time travel questions we posed on this entry. Basically, we want to understand Polchinski’s paradox, which we show in this pic So, imagine that you have a time machine. You launch a ball into it in such a way that it will come out of it one second before. And you are so evil that you prepare things so that the outcoming ball will collide with the incoming one, preventing it from entering the machine. The advantage of this paradox is that it does not involve free will, or people killing gradpas (the GPA, grandfathers protection association, has filed a complaint on the theoretical physics community, and for good reason).
No grandpas are killed, sure, but maybe the full idea of time-travel is killed by this paradox. Why should we worry? Because general relativity predicts the possibility of time-travel, and general relativity is a beautiful and well-tested physical theory. We’re worried that it might not be consistent…
There is a seminal paper by Kip Thorne and coworkers (PRD 44, 1077) which you can find here, which advances the possibility that there are no paradoxes at all… how come? In the machine described above we have focused on a trajectory which gives an inconsistent history. But there might be other similar trajectories which give consistent histories. In fact, there are infinite of them, so our problem is now which one to choose! But let us not go too fast, let us describe how would the “nice” trajectories come.
A possible alternate history: the ball travels towards the machine with speed v, but out of it comes, one second before the collision, a copy of itself with speed v’>v, in such a way that the collision does not change the direction of the initial ball (a glancing collision), but it also accelerates it… up to v’, thus closing the circle! There are no problems with conservation of energy and momentum, since the final result is a ball with speed v…
Thorne et al. described, for a case that was similar to our own, infinitely many consistent trajectories… And the question is left open: is there any configuration which gives no consistent trajectories at all? So far, none has been found, but also there is no proof for this.
And what happens when we have more than one possible consistent trajectory? My feeling is that we’re forced to go quantum! Classical physics is just an approximation. Nature, really, follows all paths, and make them interfere. But if there is a minimum action path, then it, under some conditions, may be the most important one. Quantum mechanics is happy with lots of consistent histories: they would just interfere… And a lot of funny things happen then, but let us leave that for another post…
So, what do you think? It will always be possible to find a consistent history, or not? Are there true paradoxes in time travel?
Superhighway (or funny minima)
I proposed this problem to my calculus students. It turned out to be more interesting than I thought (thanks, Ignacio and Noema).
The government intends to build a superhighway without speed limit so, therefore, without curves. It will start from city A, and should pass as close as possible to cities B and C.
In order to solve it you should start by stating what you mean by “as close as possible”. An option is to minimize the sum of the distances. But then, you get the following funny configuration:
Cities B and C are at the same distance from A, and make up a right angle. Intuition dicatates that the best route for the highway would be the bisector. Let d(A,C)=d(A,B)=1. Then, the sum of the distances from the cities to the highway is . But there is a better highway! You can just break the symmetry between B and C and make the road pass through C exactly. Then, the sum of the distances is just… 1.
If this was a real highway, the politician in charge would tell us that symmetry is also worth. Citizens of B may riot with un asymmetric solution… So, now let us change our target function. Let us minimize the sum of the squares of the distances. In that case, the bisector gives a square total distance of 1, same as the asymmetric road… Can you explain it?
Which is the best choice? Of course, it depends on the reason for which you’re fighting with the problem. If it is just to pass an exam, any of them will do…:) [Of course, there are many other alternatives. A student minimized the sum of the squares of the distances from the points to the straight line in the y direction. This makes sense in some cases, e.g.: when you’re fitting experimental points to a line.]
So, choosing the right function to minimize is crucial in practice… Kadanoff once explained in a talk that the government of the city of London has ordered a huge global study of traffic, taking into account both public and private transport, energetic and economic issues, taxes and prices, eeeeeverything. They made an enormous computer program that was running for days and, finally, told them the answer, how to optimize traffic in London. They had to remove all traffic lights. Why???? The program had many things into account… also the fact that in street car accidents, it’s normally old people who die. And old people do not pay taxes, they receive their pension money from the government. So it was convenient to remove the traffic lights… So, you see: garbage-in, garbage-out. Yes, maths is a nice girlfriend, she gives you more than you put in the relationship… but she can do no miracles. If you’re stupid, she can’t fix that…
More on compressed numbers
Spoiler alert: we are going to discuss about the previous post, and we will give the solution to the puzzle…
So, the sequence to study was 2, 12, 1112, 3112, 132112… It is easy if you read it aloud: “2” is “one two”, so “12”, which in turn is “one one, one two”, “1112”, and so on.
Let us get abstract. The transformation that takes from one sequence to the next, e.g., from 12 to 1112, is a kind of description, so we will denote it with the letter D. Transformation D is the most basic compression algorithm! Imagine that you have a sequence like “333333333”. Then, under the operation of D, it transforms to “93”, which is a neat compression. But the main lesson we get from here is that compression algorithms may expand if used without control… This trick is widely known, it appears as a puzzle for computer science students, normally starting with number 1, so: 1, 11, 21, 1211, etc. I preferred to start with 2, so that you wouldn’t find it on the web… :)
But the properties of this operator are not studied out there, as far as I know… I have been thinking a bit, and today I just made a computer program to mess around… I got a few questions for you:
- The sequence “22” is “self-descriptive”, in the sense that . Are there any other self-descriptive sequences?
- How does the length of the sequence grow? I mean: each step is increasing the length of the sequence. Is this going to stop? Is it going to grow exponentially, or what? Why?
- Can all numbers appear in the sequence ? Why?
- Let us call a sequence “primitive-1” if it comes from the expansion of a sequence of a single number, i.e.: all the , for all and . Is there any property that will tell us when a sequence is primitive-1?
I have a few hints regarding these questions, but I don’t have the answer to all of them, so we’ll play together, if you want…
Which is the next number?
If you know the answer, don’t give me the pattern. Just write the next number as a comment, ok? :)
Here is a very simple problem which appeared with my first year calculus students. Consider the function
Its domain is . But now, we may take the exponent down, using the rules of the logarithm…
and now, the domain is … What happens??? Try to explain it: (a) without complex numbers, (b) with them…