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Closed-form formulas are overrated

tl;dr A closed-form formula is a means of expressing a variable in terms of functions that we have got names for. The set of functions that we have got names for is a pure accident of human history. Thus having a closed-form formula for an object of study is also merely an accident of human history and doesn’t say anything fundamental about the object.

The essence of scientific investigation

Scientists like understanding things. A good test of understanding is the ability to predict. For example, we can claim that we have understood gravity because we can predict with amazing accuracy where the moon, for example, is going to be at any given time in the future.

In the next few paragraphs, I am going to want to make extremely general claims and that will require me to talk about some very abstract concepts. So let me talk about those abstract concepts first.

Most of the things that we have tried to understand in the history of scientific investigation can be thought of as an abstract number crunching device. The moon, for example, is something that we see in the sky at a particular angle at a given time. So we can think of the moon as a device that takes time as input and turns it into a particular position in the sky. We can denote time by a number t and the position by two numbers x and y. Thus moon converts t into x and y.

The number crunching that the moon does is not arbitrary. If one observes the moon for a while, it is easy to start seeing some patterns. Some obvious patterns are immediately visible. For example, there is a certain continuity in the way it moves, i.e., its position in the sky does not change too much in a short period of time. There are other very non-obvious patterns too. These patterns, in fact, required centuries of scientific investigation to uncover.

When we are trying to understand the moon, we are trying to understand this pattern. More precisely, we want to write down a set of rules that perform the same number crunching as the moon does, i.e., if we start with a t and apply those rules on t one by one, we get an x and a y whose values match exactly with the values that the moon’s number crunching would have given us. Now, I am not claiming that understanding the relationship between x, y and t tells us everything about the moon. Of course, it doesn’t say anything about whether there is oil on the moon’s surface. But let me just use “understanding the moon” as a metaphor in the rest of the article for understanding this specific aspect of the moon’s motion.

This is not specific to the moon, by the way. Consider some other subject of investigation. For example, the flu virus. One crude way of modelling the flu virus as a number crunching device is to say that it converts time into the expected number of people infected. That’s a very high level picture and we can make the model more informative by adding some more parameters to the input. For example, say, the average temperature that year, the humidity etc. The output can also be modified. We can, for example, make the output a vector of probabilities, where probability number i tells us how likely is it that person number i will get infected by the flu virus. There could be many ways of understanding the flu virus, but once we have asked one specific question about it, we have essentially modelled it as a number crunching device that converts some set of numbers into another set of numbers.

The main challenge of scientific investigation is that we do not usually have access to the inner workings of the number crunching device under investigation. In this sense, it is a black box. We only get to see the numbers that go in and the numbers that come out. Just by observing a large number of these input-output pairs, we take up the task of figuring out what’s going on inside the black box. We know that we have figured it out if we can replicate it, i.e., once we have constructed a set of our own rules that have the same behavior as the black box.

Things get interesting once we try to understand what kinds of rules we are allowed to write. For example, do we really have to write those rules? Is it fine if I hire a person who knows the rules and when given a time t, always outputs the correct x and y, the x and y that the moon itself would have churned out? Is it still fine if the person I have hired only understands the rules and can replicate the correct input-output behavior but can not explain the rules to me? If that is fine, then how about creating a machine, instead of hiring a person, that manifests the same input-output behavior in some way? For example, may be, the machine is simply a screen with a pointer and a dial so that when you set a specific t on the dial, the pointer moves to the correct x and y coordinates on the screen? Is that fine? Or may be, the machine is just a giant rock revolving around a bigger rock so that when a person standing on the bigger rock looks up at time t, he can see the smaller rock exactly at coordinates described by the corresponding numbers x and y?

I don’t know which of the scenarios above should be considered a “valid” understanding of the moon and which ones should not. But it seems clear that there can be several different ways of “writing” the set of rules. The primitive way of doing this was to write the set of rules as a closed-form formula.

What is a closed-form formula?

x = 2t + 1 is a closed-form formula. So is x = sin(t) + cos(t).

Until high school, I was under the impression that in order to understand the moon, one was required to present some such closed-form formula, i.e., express both x and y as functions of t. But that’s an unnatural constraint.

For example, what if x was a slightly weirder function? Say, x was 2t+1 for t < 1000 and sin(t) + cos(t) for t > 1000? May be we would still accept that, mainly because there exists a conventional way of writing such piecewise functions in math. But what if x was something even more weird? For example, say x was equal to the smallest prime factor of t? Or may be x was something that just cannot be written in one sentence? May be x was just given by a sequence of instructions based on the value of t, so that if you started with a value of t and followed those instructions one by one, you would end up with the value of x?

The punchline of this argument is that sin(t) (or even 2t+1, for that matter) is already such a set of instructions. Just because human beings, at some point, decided to give it a name doesn’t mean it is more fundamental than any other set of instructions for converting t into x. Thus in the process of understanding the moon, one should not worry about coming up with a closed-form formula.

At the same time, it is clear that some ways of writing the rules are better than others. For example, having a moon’s life-size replica revolve around the earth’s life-size replica as your set of rules is a bit inconvenient from the point of view of making predictions.

What, then, is the “correct” way of writing the rules? I want to claim that the answer to this question can be found by understanding computation and, specifically, the area of computational complexity. But I will not make this article any longer.

Euclidean Minimum Weight Matchings

The exact complexity of computing the minimum weight perfect bipartite matching in the Euclidean case is an open problem in computational geometry. This problem fits into the common theme of taking standard optimization problems on general weighted graphs and giving them a geometric flavor by forcing all the edge-weights to be Euclidean distances. Doing this often makes the problem easier to solve than the problem on general weighted graphs. Examples include minimum spanning tree (it’s open whether the Euclidean version can be done in linear time or not; the general version is known to take at least {\Omega(n\log n)} time) and the travelling salesman problem (the general version is hard to approximate, but the Euclidean case has a PTAS).

More formally, consider two sets {A} and {B} of {n} points each in the two dimensional plane. This defines a complete weighted bipartite graph where we create a node for each point in {A\cup B} and an edge {(a, b)} for all {a\in A} and {b\in B}. To each edge {(a, b)}, we assign a weight equal to the Euclidean distance between {a} and {b}. The question, then, is to compute the minimum weight perfect matching in this graph in {o(n^2)} time. Currently, the best known algorithm takes {\tilde{O}(n^2)} time where {\tilde{O}} hides logarithmic factors. If we don’t care about the accuracy, it is possible to reach almost linear time, that is, there exists a near linear time algorithm that finds a {(1+\epsilon)}-factor approximation for any {\epsilon > 0}. Getting a subquadratic approximation algorithm is a good sign because often approximation algorithms can be made exact by setting the {\epsilon} appropriately, if we know something about the solution space. For example, if we know that the set of all possible weights achievable by a perfect matching is an integer in the range {[1..n^2]}, we can get an exact solution by setting {\epsilon} to be something slightly smaller than {1/n^2}. Of course, this approach has obvious caveats, including a) that we do not know anything about the set of possible weights achievable by the perfect matchings and b) that setting {\epsilon} to be a polynomial in {1/n} will blow up the running time.

An interesting special case is when all the points are promised to belong to a {\Delta\times\Delta} integer grid. In this case an *additive* approximation algorithm is known that runs in {\tilde{O}(n^{3/2+\delta})} time, {\delta} being a small positive constant. Here the {\tilde{O}} hides logarithmic factors in {n} and {\Delta} and polynomial factors in {1/\epsilon}. From now on, we will also hide the {n^\delta} in the {\tilde{O}}.

Being on an integer grid has some advantages. For example, the weight of a perfect matching, then, is the sum of square roots of {2n} integers, each in the range {[0..\Delta]}. Sums of square roots of integers are, for many reasons, very interesting for the algorithms community and thus have been studied extensively. It is known, for example, that for any two sets of {n} integers each, the difference between the sum of square roots of the integers in one set and the sum of square roots of the integers in the other set is lower bounded by {1/f(n, \Delta)} where {f(n, \Delta)} is polynomial in {\Delta} but doubly exponential in {n}. That doesn’t quite help us yet, because setting {\epsilon} to be something doubly exponential in {n} is horrible for the running time.

In a recent paper by R. Sharathkumar, this problem was circumvented with a clever trick and a {\tilde{O}(n^{3/2})} time exact algorithm was shown for the case when points lie on a {\Delta\times\Delta} integer grid. The algorithm is really neat and works by combining a few ideas in the right way. One black box it uses is the fact that if instead of a complete bipartite graph in the two dimensional plane, you are given a planar graph, then the minimum weight perfect matching can be found using planar separators in {\tilde{O}(n^{3/2})} time. Thus his main idea is that given the complete bipartite graph, extract from it a subset of edges such that a) the subset is planar and b) it contains the minimum weight perfect matching of the complete bipartite graph. He shows that such a subset can be found in {\tilde{O}(n^{3/2})} time. To do this, he builds up on the additive approximation algorithm and uses the fact that sums of square roots of two sets of integers cannot be arbitrarily close to each other.

Being smart about distributing electricity

It turns out that the conventional way of distributing electricity is all wrong. I am talking about electricity distribution of the kind government does from the power plant to the consumers.

One of the main issues is that all the resources, the cables, the transformers, the hubs and so on, are built in order to support the peak load. But the peak load is rarely reached. In 2009, for example, 15% of the generation capacity was used less than 88 hours per year in Massachusetts. 88 hours per year! Out of the 8760 hours that a year has. Obviously, we are doing a lot of work that’s not needed.

However, we can’t really just cut down on the resources because if we do, those 88 hours of peak load will just blow everything up and we don’t want that to happen either.

Thus people have come up with an ingenious idea: control the electricity provided to the consumers such that they do not all get a large amount at the same time, thud reducing the peak load. This is done by a central hub that studies the usage pattern of different houses in the locality and schedules electricity to them accordingly. The hub can also ask the home owners to provide additional data. For example, people are usually flexible about exactly when they want to use power-consuming electric devices. So for example, the hub could ask the home owners to send a list of devices they want to use on a given day and the flexibility they are willing to accept. Next, the hub can decide the amount of electricity to provide to each house at a given time, the aim being to make sure that not many of the houses run heavy load devices at the same time.

Many other things can be done. Anything that can potentially bring down the peak load by 1-2% will save the governments a lot of money.

Building the perfect world in four extremely difficult steps

(I had written this for Goodblogs a long time ago.)

A perfect world would be one where everyone just did whatever they wanted to and lived happily doing that.

If tomorrow, everyone just decides to do whatever they want to, the world will turn into a chaos. For example, no one will want to clean the garbage and as a result, we will eventually rot in our own filth. To keep the world working, it is necessary that at least some people do things that they don’t particularly like doing. Can we repair this? In general, can we design a perfect world, a world where you never have to feel guilty, a world where you can just do whatever you feel like at any given moment and that will be the best thing to do for you and for the society? If yes, then what are the steps we need to take?

To understand this, we first need to understand what is the best thing for the society. There are things that are good for some people and bad for the others and there are things, that are good for the society right now, but in the long run, will lead to the decay of mankind. Let’s, for the time being, define ‘best’ as the thing that has the best average over people and over time. That is, we take the average happiness level of the world at this time and then take the average of this average over time. The best thing then, would be the thing that maximizes this average.

Once we have this out of our way, we can understand that the essential problem is to align what an individual feels like doing at a given moment and what’s the best thing to do for the society at that moment. Since our definition of ‘best’ depends on happiness levels of people, there are two extreme approaches to solving this problem. One extreme is to reprogramme the human brain so that it feels happy or sad in a more controlled way. This extreme is slightly trivial. All we need to do is to build the perfect mood enhancing drug and make it compulsory for everyone to take it. This will suddenly boost up the total happiness level of the world. The other extreme is to leave the human brain untouched and reengineer the world in such a way that whatever we want at this moment is made possible immediately. This is perhaps impossible. It’s easy to imagine an individual getting so angry at another person that he genuinely wants to kill him, or harm him severely in some other way. If then, this were made possible immediately for him, it would create more grief to the person being harmed than happiness to the person inflicting the harm. It’s similarly easy to imagine completely outrageous, or even physically impossible wishes that a person can make. One might have to break some laws of physcis to make that possible immediately. Since this approach seems impossible and the other extreme is kind of sad, the ideal should lie somewhere between the two extremes.

One possible midway approach is the following.

Step 1 – Build robots to do all the dirty work that no one in the world wants to do but is necessary to be done. This will leave out the kinds of work that at least some people in the world like doing. Let them do that work. But there might still be problems. The person who likes doing the work X might be living in Japan and the place where X needs to be done might be in Canada. Moreover, if we consider one person who likes doing X, then he may not want to do X all the time. The time when X needs to be done in Canada, he might be in a mood to go swimming with his kids.

Step 2 – Build a global work organizer. This will be some huge global device that will take the help of the internet. It will monitor what each person in the world is in the mood of doing right now and the things that need to be done at this moment in different parts of the world. Then, it will match the tasks to the suitable people. Since the world is such a large place, we can assume that what a random person wants to do at a given moment is useful for someone somewhere in the world and what’s useful for a given person is being wanted to be done by someone somewhere in the world. If there is some task, where this doesn’t happen, we already have step 1 to take care of it. Such things are already being done. There are several outsourcing services online for tasks that do not require physical presence. For example, Mechanical Turk and oDesk are websites that are designed exactly for this purpose. Even GoodBlogs is similar. The previous post I wrote was originally intended to be an email to my mother. But somehow somewhere in the world, there was a group of people who agreed to give me $20 for it. However, building this global work organizer will still not build the perfect world. For that, we will need the next step.

Step 3 – Upgrade the human mind so that its emotions are in control. For example, no one should ever feel like seriously inflicting any harm on anyone else. Not just this, but one will also need to make sure to not inflict any harm on the future self. Even if everyone is full of kindness and love towards others, one still might want to smoke a cigarette and thus suffer with lung cancer later in life. If not, then one may simultaneously want to learn how to play the piano and to not practise, or, to get a girl and to not develop the skills to woo women etc etc.

Step 4 – Make learning easy. Even if we are the masters of our own emotions, have robot servants and are never forced to do something that we don’t want to do, there might be a situation where we want to learn something quickly but we can’t. For example, one one may be craving good food, but not know how to cook. Then it will be good to have a plugin that they can install on their mind to give them that feature in a few seconds.

Intelligence and How to Get it

Some time ago, I read a book called Intelligence and How to Get it: Why Schools and Cultures Count, written by Richard E. Nisbett.

That’s when I became intelligent.

The book has some very interesting research about intelligence. It basically tries to argue that intelligence is not completely in the genes.

Anyway, while I was reading the book, I started writing up some interesting stuff that I read, and emailing it to some friends. The idea was of course, to archive things so that, you know, when 10 years from now, I start writing my New York Times bestseller, I have access to all the material I was reading.

Now, I realized that the emails can actually be shared online. So here is a pdf that has all the emails in this series. When I was writing those emails, I was not trying to make them into well organized essays. Thus this pdf looks like a collection of various independent thoughts. I have separated those indpendent thoughts with a line made of asterisks.

I did the same thing with some other books that I read too. I will post their respective pdf’s here as well.

The New X-Men Movie

It’s annoying when someone who has never taken a Physics 101 course tries to make a sci-fi movie. Case in point is the new X-men movie – X-men first class. Let me explain.

My first problem is with Magneto. He is the superhero who can change himself into a magnet and pull or push things that are made of iron, or other magnetic materials at his will. All that is acceptable. What’s not acceptable is when he pulls a whole submarine out of the ocean. Even if we assume that somehow he is able to muster enough magnetic strength into his body to significantly move such a heavy object against the strong pull of gravitation, my problem is this – what happened to Newton’s third law of motion?

We were taught in high school that forces exist only in pairs called action-reaction pairs. That is to say, you cannot apply a force on another object independently. Whenever you try to do that, the other object will apply an equal and opposite force on you. For example, when the earth pulls us down, we pull the earth up with the same force. It’s just that since our mass is so insignificant compared to the mass of the earth, the force has a visible effect on us but very close to zero effect on the earth. And since we are distributed all over the surface, these already negligible effects cancel each other out to a large extent.

Now if we look at Magneto pulling the submarine out of the ocean with the sheer pull of his magnetic body, Newton’s third law would suggest that the submarine should also pull Magneto downwards with the same force. And since Magneto’s own mass is negligible compared to the mass of the submarine, the effect that you see on the submarine should be negligible compared to the effect you see on Magneto. What I am trying to say is that if Magneto can apply a force enough to pull the submarine out, he himself should fly at an enormous velocity towards the submarine and die within milliseconds of the impact.

You might argue that come on, he’s a superhero and so he can do everything. But the problem with this argument is that he actually has clearly defined powers. His power is to turn himself into a magnet. Honestly, if a superhero could violate Newton’s third law, then this power would be way more impressive than being able to turn himself into a magnet. He should have been called the third law violato or something, instead of Magneto.

My second problem is that dude who could fly simply by wearing wing-shaped clothes and screaming in an ultrasonic voice. In fact, I can’t fathom a reasonable scenario that could lead someone into thinking that these two things were related. The only thing that comes to mind is this – bats emit ultrasonic sounds and they can fly. But really? Is that all that you need to be convinced that you can fly by producing ultrasonic sounds? Human history is full of incidences where someone died because of an attempt at flying without thinking things through. Is this really all they were missing? Scream at the top of your voice while flying? I wonder why inspite of so much advancement in flying technologies, there does not exist a single aircraft that works on this ultra-sonic sound princple. It should be way easier than all that stuff the Wright brothers did! All we really need is a loose sweater and a sonographic equipment from the nearest hospital!

Textbook that teaches how to swim

I have been blogging at GoodBlogs. It’s a nice new website that gives you money if your blogs get upvoted to the main page. Here’s my most recent blogpost – http://www.goodblogs.com/view-post/Texbook-that-teaches-how-to-swim .

And here’s the text pasted from the original post –

Learning American history is different from learning how to swim. You can learn American history by reading a textbook about it, but I don’t think there exists someone who read a textbook on swimming, jumped into water and started to swim. Why are some skills so different from others in this respect? Wouldn’t it be awesome if all skills in the world could be learned just by reading a textbook? Can anything be done so that this does happen?

American history can be learned by reading a book because it mostly consists of facts and they can be easily described in a book. So any skill that consists of a collection of information can be transfered by encoding the information in some form (in this case, a book) and passing it on. Is swimming a collection of information too? If you think about it, it actually is. If you study in detail exactly what a swimmer does and exactly why he is able to swim, you will realize that his body is following certain rules. For example, if you feel some flow of water in this direction, then push water away in this direction, or, fold your legs a bit at the knees, then apply some force with the muscles in your thighs etc. Indeed, the brain is finally an information processing device. All it does is that it gets information from different sensory organs, uses it to decide upon an action and sends instructions to different parts of the body so that the action is executed. Whatever information processing the brain does while swimming is also just a piece of information, and, in principle, can be written down in the form of a book and transfered to others. But people learned swimming way before they understood the exact mechanism with which we swim. If you talk to an expert swimmer, he will most probably not know the exact mechanism either. He just somehow gets it. For him, the rules are all of the form, “when you feel this way, you should move these muscles this way,” where the ‘this’ are vaguely defined somewhere deep inside his brain.

So can a texbook for swimming be written? In principle, yes. You just describe the precise set of rules that your brain follows in excruciating details including the exact muscles it moves by exactly what amount when exactly what happens to the water around you. But this textbook will hardly be useful. If you read this textbook and memorize all the rules and jump into water, by the time you understand the force of the water stream and try to decide which muscles you are going to move, you will already be twenty meters deep into the water surprised about the fact that you haven’t had any oxygen in a long while now.

There are other skills that are similar to swimming in this sense. For example, consider learning to play the piano. Even Bach’s Toccata and Fugue in D minor can be written down on a piece of paper using weird symbols that encode the sequence in which you have to press the keys and for how long and how hard and so on. In principle, you could just pick up the staff notation and go through it and you would know exactly how to play the song. But once again, once you begin to understand what’s the next key to press, the audience will probably already lose track of which song you are playing and will start leaving for home.

The thing that’s common with these kinds of skills is that even though writing a textbook about them is useless, if you are allowed to slow time down, it might still be possible to learn the skill just by reading the textbook. For example, in the case of swimming, if somehow the laws of physics slowed down and let you think and calculate your next moves before applying gravity on you, you would be able to swim just by reading the textbook. Also, if the audience were a bit patient and did not lose track of the song so easily, it would be possible for you to just read the staff notation and play the song without any practice.

But now consider cooking. Suppose you want to learn enough cooking so that you can design recipes that you like. Can a textbook be written that gives you exactly this skill? The textbook will once again need to specify some rules in detail. For example, what exactly happens when you add ginger to a recipe? How exactly does it change the taste? Or what happens when you bake something in the oven instead of deep frying? The problem with these questions is that the answer just cannot be specified in a precise way. For example, what exactly is the difference between deep frying and baking? How do you explain this to someone who has never tasted food that’s deep fried or baked? How do you explain to someone the taste of anything if that person has not had that thing (or something similar) before? You can’t. You have to tell him that here, this is baked and this one is deep fried; taste and find out the difference. All other methods are just vague and not sufficient to teach cooking to someone. So in this sense cooking is different from both learning American history and learning how to swim.

So this seems to define three different categories of learnable skills – 1. those that can be taught through textbooks (eg., American history), 2. those that can be taught through textbooks only if one is allowed to slow down time (eg., swimming) and 3. those that just cannot be taught through textbooks (cooking).

Reducing the category number of a skill will be a good progress for science. For example, bringing some skill from category 3 to category 2 will be quite cool. How would one do that with, say, cooking? Well, we will need to make things precise. We will somehow need to quantify the different tastes and figure out exactly how much of which taste is contained in what ingredient. Then, a typical textbook on cooking will say things like, “one cubic centimeter of turmeric contains 3 units of spiciness, 4 units of something and 2 units of something else.” There will be tests for understanding what kinds of tastes you like. Then, whenever you want to cook something delicious, you will have to look at the book for measurements and pick something that brings the taste into the range that you like.

Bringing a skill from category 2 down to category 1 will be quite different. Well, if we are allowed to cheat by using machines, then we are already doing that. We have machines that can swim. We have submarines that can take us inside water and follow our instructions about where we want to go. And to teach these machines, one only needs to give them some information in the form of computer programs. If we are not allowed to cheat, then one will have to figure out some way so that our brains are able to do all the calculations really fast, so that we can decide which muscles to move in real time. We will somehow need to augment our brains, perhaps by building machines inside it, if not outside.

The Fun Theory

Obeying traffic rules, throwing garbage in the garbage can, recycling stuff, not making too much noise in a residential area and switching off electrical appliances when not using them are all examples of things that will significantly improve human condition if everyone in the world starts following them. However, not many people do, even the ones who are convinced that following them will improve human condition in the long run. The reason is that we are all programmed by evolution to not care about the long run. We are all victims of hyperbolic discount. We take most of our decisions based on what will give us immediate rewards and what won’t. Next time you find yourself swearing at a person who’s driving like an idiot, think about all the hours of your life you have wasted procrastinating. You are very similar to that person in the sense that both of you are doing something that you shouldn’t be doing just because your brain isn’t capable enough to consider the big picture every time it’s making a decision.

 

So what’s the solution? You might think that the solution is simple. All that’s needed is that people should become intelligent enough to understand what they should do and then do that and that it’s a pity that they are not that intelligent. Unfortunately, this is not as much a solution as it is stating the problem in a different way, in the sense that solving this is as difficult as solving the original problem. If this is all you have had to say, then you have made no contribution. A real solution is to accept the fact that people’s (including your own) brains are limited and then design a system where people somehow get immediate rewards on doing good things. It’s not necessary to make the rewards monetory. It can be anything, even a simple pleasure. This is exactly what’s being done at http://www.thefuntheory.com/ and it’s totally awesome. I found out about them just now and I suddenly have too much respect for them.

Minimum enclosing ball

Computational geometry is full of algorithms that demonstrate how randomization very often leads to simplification. One example is the famous randomized incremental algorithm for finding the minimum enclosing ball of a set of points in d-dimensions. The idea of the algorithm is very simple. All you need to do is to look at the points in a random order and maintain the minimum enclosing ball (MEB) of the points seen till now. When a new point is considered, if it already lies inside the current MEB, then there is no need to do anything, but if it lies outside, then update. With some math it can be shown that the expected amount of time this algorithm will take is linear. This is at par with the best known deterministic algorithm known at the time of discovery of this algorithm. However, the deterministic algorithm was quite complex and not easily implementable.

Here’s a very nicely written paper that describes the algorithm – http://www.springerlink.com/content/ph273841264x5163/

Horizontal and vertical

Let’s say you are blindfolded and taken into a gravity free spherical room where you lose all sense of direction and then you are presented with two square shaped sheets of metal. In addition, it’s promised that if a person standing outside the spherical room looks at the sheets in their present positions, then he will say that one of them is horizontal (meaning it’s lying down) and the other is vertical (meaning it’s standing up on one of its edges). Next, you are given the permission to take those sheets in your hands and inspect them in whatever way you want. Your task is to somehow decide which one was horizontal and which one was vertical when they were given to you. To help you out, you have been connected on phone to a person who is standing outside the sphere and who can see the two sheets through a small whole. You are allowed to ask him as many questions as you want, but the only problem is that he speaks French. So if you point to a sheet and ask him if it is vertical or horizontal, he will tell you the right answer, but you won’t understand it. Can you still accomplish your task? (I am assuming you do not understand French.)

I just came up with this puzzle a few minutes ago. I want to know if the answer I have in mind is trivial or not.