2009-09-18

Text-Chatting Aliens

The aliens flying above Bengt's house are having some difficulties contacting their home planet. They want a unified protocol for sending information about location from each of their many space ships scattered around the galaxy, and they want to use earthly communication standards, so as not to be detected. They tried using TCP/IP, but found it to be a terrible and outdated standard. For lack of a better alternative, they decided to use SMS.
Each text message can contain either the name of a star (the aliens all know where the stars are), or a number (such as the distance from said star). They want to use as few messages as possible.

    Device a system for them to identify a location. The aliens will all be informed of the new standard. How many messages will they need?

2009-09-10

Phone Strategy

Bengt is doing his exercise routine. It takes quite a while, mostly because he is too lazy to work very hard, so he doesn't get tired very quickly. In fact, his so called exercises mostly consist of posing, because he has seen that body builders do that a lot. It takes a predetermined time before Bengt gets tired, but there is no way to know in advance how long that is.
For some reason, Bengt has several friends. Now Sture wants to see him, but he knows that he has to wait until Bengt is finished with his exercise routine. It wouldn't do to disturb Bengt in person while he's busy with something so important, so Sture will phone him instead. The more often he calls, the sooner he will find out that Bengt has finished. Unfortunately, whenever he calls, Bengt has to take a break for one minute to answer the phone, so the time until he is finished is delayed by one minute. Sture now has to come up with the best strategy for getting a hold of Bengt as soon as possible.

  1. Suppose Sture knows that Bengt always exercises for an hour, but he doesn't know how long it has been since he started. At which times should Sture call?
  2. Suppose instead that Sture does know when Bengt started, but that the exercise varies. Sture has determined from experience that the exercising takes one hour on average, and that the probability that he will finish in the next short moment remains constant. If he doesn't call, after how long is the probability 1/2 that Bengt has finished?
  3. With those conditions, at which times should he call?
  4. Suppose instead that the probability density of finishing decreases, so that after a time t (not including phone pauses) the probability that he is still exercising is 1/2^t. What is then the probability density?
  5. In that case, when should Sture call?
  6. Suppose instead that Sture has already determined that the best times to call is once every hour. What is the probability density?
  7. And suppose it is instead best to call after 2^n minutes, for non-negative integer n. What is the probability density?
  8. After all, Bengt has more than one friend. With the conditions in question a, suppose Alban is also trying to call Bengt. Neither knows when Bengt started, nor when the other person called. Each wants to reach Bengt as soon as possible. As soon as one of them reaches Bengt, the other person has to wait another time p before Bengt is finished with the first person. What is the optimal phone strategy this time?
  9. And what if we do the same thing with the conditions from question b?

2009-07-20

Marble Problem 3

Sture has finally got tired of playing with marbles, but Bengt continues with the fervor of a true scientist. He is still building pyramids, of four marbles each. But the floor can be a little slippery sometimes, and then the pyramids won't stick together. Bengt intends to do something about that. One of his ideas is to give the marbles electrical charge, in order to make them stick to each other.
The marbles have diameter 1 cm and mass 1 g. They are not conductive, so they do not lose their charge when touching each other or the floor.

  1. An easier way to get marbles to stick together is to use glue. Suppose each marble exerts a glue-force on each marble that it touches. How big must the glue-force be to hold the pyramids up, if there is no friction at all? (That is, neither between a marble and the floor, nor between a marble and another marble.)
  2. Forget the glue, and let's investigate the friction a little more. What happens if there is a large friction between a marble and the floor, but no friction between a marble and another marble?
  3. And what if it's the other way around - large friction between a marble and another marble, but no friction between a marble and the floor?
  4. Suppose the friction against the floor is very large. How big must the the marble-marble friction coefficient be to keep the pyramid up?
  5. Suppose both the frictions are the same. How big must the friction coefficient be to keep the pyramid up?
  6. Bengt also wants to build the ball-shape mentioned in Marble Problem 1: One central marble, surrounded by as many other marbles as possible, all touching the central marble. With no friction, what glue-force is needed to keep this arrangement standing?
  7. Without glue, could enough friction keep it up? Assume very large friction against the floor. How big must the the marble-marble friction coefficient be?
  8. Now suppose we want to do something even more extreme: Let there be a force between two marbles, which is just enough to overcome gravity, so that Bengt can hold one marble and let the other hang from it. Clearly this is possible with a glue force. But could this force be described by a friction coefficient, if it is extremely large?
  9. Let's try with those electric charges. Two marbles have equal and opposite charge, enough to hang one marble from the other. How big must the charge be?
  10. What if you have three marbles hanging like a chain, the one in the middle being positive, and the others having equal but negative charge?
  11. What if you have n marbles in a chain, with equal and alternating charges?
  12. Returning to the case with three marbles - is there a better way to distribute the charges? Find the setup such that the sum of the absolute values of the charges is as small as possible.
  13. And for n.
  14. We can also use charges to get the pyramid to stay up. If the top marble has positive charge, and the bottom ones have negative charge of the same magnitude, will that help the pyramid stay together, or will it actually force it apart?
  15. Suppose instead that the charge of the top one can be larger than the others. What should the charges be to get the pyramid to stay up? Assume there is large floor friction but no marble-marble friction, and minimise the largest charge magnitude.
  16. Even more freedom: Arrange the charges, magnitude and polarity, any way you like, and again minimise the largest magnitude.
  17. And what if there is no friction at all?
  18. What about the ball shape? If there is more than one possible shape, which one is better? Assume that there is large floor friction but no marble-marble friction. Also assume for convenience that the middle marble is positively charged. To get the best result (that is, minimise the largest magnitude), which marbles should be positive and which should be negative?
  19. And what would those charges be?
  20. And what if there is no friction at all?
  21. There is yet another, even more unrealistic way to get the marbles to stick together. What if they were really heavy, so that gravity kept them together? Consider the case of one marble hanging from another. If they have the same mass, how heavy must they be?
  22. And what about the pyramid? Assuming large floor friction but no marble-marble friction, and same mass for all marbles.
  23. And what if there is no friction at all?
  24. For completeness: The ball shape. Large floor friction but no marble-marble friction.
  25. And what if there is no friction at all?
  26. Bengt decides to finish off the alphabet by putting u and i together. Take one of the marbles from question u, charge it up, and hang it from a normal 1 g marble like in question i. How big would the charge have to be? And how big, roughly, is the theoretically biggest positive charge 1 g of normal material can have (that is, with every electron removed)?

2009-07-13

Marble problem 2

When we left them last week, Bengt and Sture were building pyramids out of marbles. Since time in problem-space is independent of time in the real world (and also, I wrote this problem a while ago) they are still working on that, but they are nearly finished. Oh boy, they have built lots of pyramids.

There are twelve marbles left in the bag. Each marble is either black or white. The marbles are otherwise identical. Sture takes four marbles, and then gives the bag to Bengt. Bengt takes three marbles and places them on the ground. They are all white. He reaches in to take one more.

"I wonder what the probability is of the last one being white." says Bengt.
"Well, if those three were white, then there are most likely fewer white left."
"But on the other hand, the fact that these three were white suggests that there were more white ones than black ones when you gave me the bag."
"Hm. Sounds like a good problem."
"I don't know. I think it might be a bad problem, actually."

  1. Is it a good problem? Or, more specifically, is it a problem that can be solved? Is it possible to say anything about the probability with the given information, and in that case, what?
  2. Assume that half of the marbles are white when Bengt gets the bag. Same question.
  3. Assume instead that half of the marbles are white to begin with, and that Sture takes four at random.
  4. Assume instead that half of them are white to begin with, but Sture takes four of the same colour.
  5. Assume instead that half of them are white to begin with, but Sture starts by taking two of the same colour, and then picks the other two at random.
  6. Assume instead that we don't know the original distribution, but that each marble is originally equally likely to be black or white, and that Sture takes four at random.
  7. The same, but Sture takes two of each colour.
  8. The same, but Sture takes four of the most prevalent colour. (If they are equally common he picks a colour at random.)

2009-07-06

Marble problem 1

Bengt so enjoyed having a themed month, that he insists on doing it again. This time it's Marble Month, so he invites Sture over to play with marbles. Sture says that he has lost his marbles, but that he'd be willing to make trite jokes about balls instead. But Bengt has made up his mind, and we all know how stubborn he can be.

Bengt brings out his bag of marbles. The marbles are physically identical; they have the diameter 1 cm, and mass 1 g. They start building pyramids, by placing three marbles next to each other in a triangle, and one on top.

  1. With four marbles you can build a pyramid of 2 levels; how many marbles do you need for n levels?
  2. And how high is such a pyramid?
  3. Another shape that one can imagine, altho it's hard to build with marbles, is a shape where a central marble is surrounded by as many other marbles as possible, each touching the central one. How many marbles can you fit in?
  4. In how many shapes can it be done?
  5. That arrangement can also be extended. You can add another layer, such that each marble in the new layer has to touch one of the last layer. If you keep going, the shape will look more and more like a regular polyhedron. Which one?
  6. How many marbles will you have in n layers?
  7. Imagine that the pyramid is enclosed in a tetrahedron. It is just big enough to contain the whole pyramid. The walls are water- and airtight, but have negligible mass and thickness. Would the contraption float on water?
  8. They build a number of pyramids - the small kind, just four marbles. If a pyramid "goes off", the middle marble drops down and the other three roll away at the same speed. What speed will they move at?
  9. In the end, they will have n pyramids. They are randomly distributed in a roughly circular area a of the floor (a really big area), and oriented in random directions. If a marble hits a pyramid, it will go off. It should therefore be possible to set up a chain reaction of the pyramids, with marbles from each busted pyramid setting off other pyramids. How big must n be to achieve "critical mass", so that setting off a few of the pyramids is likely to set off a large part of the others? Assume that there is no friction, and no energy loss in collisions, and that the marbles disappear when they leave the area.

2009-06-08

Bad Traffic Behaviour

Bengt is walking along an infinite straight road. He has 1000 s left to walk. But he also needs to cross the road. The safest way to cross is of course to go straight across, perpendicular to the road, which would take another 10 s. But Bengt is determined to cross the road the fastest way possible. What makes it more complicated is that there are cars arriving at random. The average frequency of cars is one per 100 s, and he can spot them 10 s in advance. To a good approximation, the cars are moving infinitely fast (so Bengt's movement does not affect the time it takes the car to reach him), they are very wide (so Bengt has to be completely across before any car reaches him) and there is no correlation between their arrivals.
So he wants to figure out the right path to follow in order to minimise the expectation value of the time. As soon as he sees a car, the answer is obviously a straight line from his current location, such that he reaches the other side in 10 s. At the moment he can't see any cars, but they could appear at any time.

    Which path should he set out to take across the road?

2009-06-01

Ill Logic

Bengt is feeling ill, and is lying in his bed playing around with ternary logic, modal logic, and some other things that he's not sure about the proper terms for.
Given these premises:
  • All donks are ganks.
  • Every gank was created by another (older) gank.
  • Binks are more likely to be donks (in other words, there is a higher fraction of donks among binks than among non-binks).

Evaluate the following statements as definitely true, definitely false, or unknown.
  1. Donks are more likely to be binks.
  2. Some ganks are donks.
  3. Some things are not donks.
  4. There was a first gank.
  5. There is an oldest gank.
  6. If some things, but not all, are binks, and there are no ganks, then some donks are not ganks after all.
  7. If all things are binks, then there is at least one donk.
  8. If all things are binks, and there are some things that are not ganks, then there is at least one donk.

2009-05-24

Chess problem 3

Chess Month continues with another very chaotic chess challenge.

  1. What should white do?
  2. Bengt has come up with another peculiar game. It's rather zen-like, or so Bengt thinks, anyway. Others might say "boring" is a more fitting description. Anyway. There are three pieces, all identical. The board is a chessboard, and it is empty when the game starts. In each turn, the player can add a piece to the board, or remove one, or move one from any square to any other empty square. You are not allowed to make a move so that you get a situation on the board which you have had before. (A different piece in the same square counts as the same situation. You can't add a piece after it has been removed.)
    How many turns can the game last, at the most?
  3. This game could also be played with two players. If the winner is the first person who is not able to make a move, who wins - the player who starts, or the other one? (Assume that they are using optimal strategies, that they want to win, and so on.)
  4. How many turns will it last?
  5. Do questions b, c and d again, but with n pieces instead.

2009-05-15

Chess problem 2

Bengt doesn't often play chess. It's not that he doesn't like it, but he gets upset when he loses, and people don't like Bengt when he's upset. So most of his friends don't play with him anymore. But he still likes to make up weird chess problems, usually involving completely unrealistic situations, that would never appear in an actual game. Here's another one:

  1. What should white do? Bengt can come up with a rather simple solution that leads to mate in four moves, and a more unusual one that leads to mate in only two moves.
  2. Since Bengt doesn't have anyone to play with, he makes up games and challenges for himself. His latest idea is this: You place a number of white pawns in any of the 16 squares forming one quadrant of the board, let's say the upper left. Then you place a number of black pawns in the quadrant diagonally opposite, in this case the lower right. When you've placed all the pawns, you suspend the board, balancing it on the diagonal between the quadrants, so in this case you lift the lower left and the upper right corner, so the board can pivot around that diagonal. The aim of the game is to get as many more black pawns than white pawns as possible (black pawns minus white pawns should be as big as possible), but so that the board does not tip over to the black side. How many is it possible to get?
    Assume that the pawns are pointlike and stand in the middle of the square. You are allowed to use more than one chess set, that it, more than eight of each type of pawn.

2009-05-08

Chess problem 1

There are many kinds of good problems, not just ones involving complicated maths and physics calculations. So, this month, it's Chess Month! Here's the first one:


  1. What should white do?
  2. We can't avoid adding a little bit of maths after all. Think of the chessboard as a mathematical pattern. Each point on the board is either black or white. The colours are equivalent; in other words, if you invert the colours and rotate the board, you get the same thing. The length of the side of a square is 1, and there are no points outside the squares. Now, imagine drawing a line so that every point on the line is white. What is the upper limit for the length of such a line?
  3. Is there any maximum length at all?

2009-03-12

Too Fast for Physics

Bengt and Sture are cruising down the motorway. Bengt is driving.
"Why are you driving so slowly?" says Sture.
"I'm not driving slowly."
"Oh, I get it. You're going to say that since you're driving east, the rotation of the Earth is added to our velocity, so we're actually going really fast."
"No, that would be silly. Also, in this place at this time of day, the rotation of the Earth around its axis is completely counteracted by the rotation around the sun, so the speedometer shows our actual speed in the inertial reference frame of the sun. No, I'm just saying I'm driving as fast as the speed limits allow."
"Which is really slow. QED. And by QED I mean 'quick easy driving'. I never thought of you as the kind to follow laws."
"I always follow the laws of physics."
"Yeah, and I'm not asking you to drive faster than 3*10^8 m/s. Just a little bit quicker than this."
"Well, I'm just being an environmentalist. Mostly because it contains the word 'mentalist'. You know, like a mind-reader or something."
"What do you do when people find out that you can't read their minds?"
"I just claim that I meant that I'm part of the philosophical school which is also called 'mentalism'. And I look at them as if they were really stupid."
"Ooh, good one."
"Thanks. Anyway, speaking of air friction..."
"We were?"
"Yes: with higher speed, the air friction is higher, which increases the fuel consumption, which is bad for the environment, which is why environmentalists don't drive fast. Try to keep up."
"Right."
"So I was wondering: How much does the air friction increase with the velocity? In first approximation, it's probably a power of the velocity. So is it v, v^2, v^3, or what? If you drive twice as fast, is the air friction twice as big, four times as big, eight times as big?"
"Hm, good question. I think we can solve that with dimension analysis. The force should reasonably be directly proportional to the density of the air, and to the area of the front of the car, right? Then we'll see which power of v fits, to give the dimension of force."
"Sure, but there's a quicker way. You hit twice as many air molecules, and they travel twice as fast, so it should be at least v^2, maybe v^3, but hardly more."
"Slightly vague, but okay. How do you know which one?"
"It has to be v^3. Because v^2 is a scalar. If it was v^2, you would get the same result if the car was going backwards. Replacing v with -v, we get the same force, but it should be negated."
"Good point. But anyway, you drive too slow. Next time I want to drive."
"Then you'll have to stay sober."
"Oh."
Sture looks disappointed. They sit quietly for a while, thinking. It's hard to tell whether they're thinking about air friction or about beer.
"You know, it's probably not good for traffic safety, doing physics while driving. You know what they say, 'don't drink and derive'." says Sture.
"Yeah, but we always do that anyway, don't we? And this would just be a case of drive and derive, except you don't need derivatives to do dimension analysis, so your whole point is moo."
"You mean 'moot'?"
"No, I mean 'moo'. I mean it's really stupid, like something a cow would say."
"Or we can take the train. Trains are good." says Sture.
  1. What result does the dimension analysis give?
  2. If either of the arguments was wrong, where did it go wrong?
  3. Bengt claims that the rotation of the Earth around its axis is completely counteracted by the rotation around the sun. Ignoring the actual speeds involved, what time of day would it have to be?
  4. Not ignoring the speeds, is it actually possible?
  5. If it is: How far from the equator are they? If not: How many times bigger would the planet have to be to make it possible?
  6. If you are at the equator, how fast would you have to go to stand still in the sun's reference frame?

2009-03-10

Mystery Wheels

Bengt has made a new invention. It's a randomiser.
It has a number of wheels, wheels which are partially conductive. The wheels spin, and there is a contact brushing against each wheel. All the wheels are connected in one circuit, so each wheel can either let the current pass or break the circuit. It acts as a switch, switching on and off once per turn.
Then there is a little device which sends a short electrical pulse through the circuit and registers whether it makes it through or not. There are n wheels, and the fraction of the wheel which is conductive is 1/(nth root of 2), so the probability of the pulse going through is 1/2.
The wheels all look the same, but they spin at different (constant) speeds. Bengt figures that since the ratios between the speeds are not rational numbers, the same configuration of wheels will never occur twice, making the device a really great randomiser - you will never be able to predict the result of a pulse. That's the idea, anyway.
One way to use the machine is to send pulses periodically, with a certain frequency, to get a randomised bitstring. This frequency can be assumed to be significantly lower than the rotation speed of the wheels.
  1. Will you eventually be able to guess the next bit in the bitstring, that is, the outcome of the next pulse? Will you be able to guess it only with a limited degree of certainty, or will you ever be able to predict it with 100% accuracy?
  2. If it is possible to reach enough predictability to make the randomiser unreliable - that is, the number of correct guesses become significantly larger than the number of false guesses - how does the time it takes to get there depend on the number of wheels?
  3. How does it depend on the speed of the wheels? Will faster wheels make the randomiser better? How much better?
  4. Is it possible to calculate an actual upper bound for how long it takes to reach predictability, as a function of the number of wheels, the upper bound of the speeds of the wheels, and the pulse frequency? If it's possible, please do.
  5. Suppose you are capable of sending pulses at any time, as long as there is a certain minimum delay between each pulse. Sending pulses periodically with exactly that delay will give you a certain bitstring. Now, in order to predict that bitstring, is there any other pattern of pulses that would do it faster? That is, is there any pattern of pulses that would give you the information needed to predict the periodic pulses faster than the periodic pulses themselves?
  6. Provided you have the added advantage of knowing the outcome of the previous pulses, is there any faster way of "cracking" the machine than sending pulses at a predetermined pattern? (No, you're not allowed to physically crack it open.)

2009-02-13

Numbers that Grow on Trees

Bengt has come up with a whole new kind of number system, which he has decided to call "Dendriux". Each number is expressed as a full binary tree - that is, a tree diagram where each node has either two child nodes or none. It's also binary in a different sense; each node contains either a 1 or a 0.
The numerical value is calculated as follows:
- A node with a 1 has a value which is 1 less than it otherwise would be.
- A (0) node with no children has the value 2.
- A (0) node with children of value n and m has the value of the sum of the nth and the mth prime number (counting 2 as prime number number one).
  1. Some numbers can be written more than one way. Find, for example, all ways of writing 10, 11 and 12.
  2. Suppose that f(x) is the number of ways in which the number x can be written. Find the asymptotic upper bound (the "ordo") for f(x).
  3. (hard) Is it really possible to write all positive integers this way? Prove it.

2009-02-05

Nightly Tag

Alban is visiting Bengt, and after some drinking they start getting childish and decide to play tag. Bengt is surprisingly fast for his size; more specifically, he is k times faster than Alban. They are both moving at constant speed.
Bengt is chasing Alban, and they are a distance d apart. Then suddenly the sun sets, and Bengt can no longer see more than 1 m ahead.
  1. Suppose Alban is performing a brownian motion. What curve should Bengt follow in order to catch him as quickly as possible?
  2. For which values of k and d is it likely that Bengt will catch Alban?
  3. Now suppose Alban can follow any curve. What curve should Bengt follow to be sure of catching him?
  4. For which values of k and d can Bengt be sure of catching him?
  5. Eventually they get tired of the game. They lie down it the grass and start idly chatting about spaceships.
    Repeat all the above questions, but now assuming that Bengt and Alban are in space; that is, they move in three dimensions rather than two, they have a constant (magnitude of) acceleration instead of constant speed, with Bengt accelerating k times faster than Alban, and their initial relative speed is s.

2009-01-22

More Solid Geometry and Booze

Bengt is having a birthday party and wants to offer his guests something really special. He discusses with Sture what kind of drink you could give the guests to impress everyone.
"It has to be something that no one has experienced before." says Bengt. "Were you at Percy's party last week?"
"No, what drink did he have?" asks Sture.
"A 'Bloody Mormon'; one part vodka, one part juniper berry and one part meerkat blood."
"That's not very imaginative. I had one of those the already in 1963."
"But hey, Gilbert had a really interesting drink at his party last year. A saturated solution of salmiac (NH4Cl) in water, served in half-spherical glasses with radius 5 cm."
"Yeah, it had quite a bite. That won't be easy to beat."
"Ah, don't worry. You don't play in the top league unless you're good. I think I have an idea, actually. What are the two most important properties of a good drink?"
"Strong and cold."
"Exactly. So how does this sound: More than a hundred percent alcohol, and a serving temperature of minus 104 degrees?
"That sounds absolutely marvellous."
"I was planning to use some rather special cups to measure the drinks. I have ten canisters, shaped as each of the platonic bodies, five with the side one dm and five with the side √10 dm. The thickness of the walls of the canisters is negligible. So you use them to measure the drinks, and then you pour them in big glasses."
"And then you pour them in your own non-platonic body."
"Exactly."
  1. Bengt would like to be able to measure 1, 1/2, 1/3, 1/4 and 1/5 liters, using only completely filled canisters. Is it possible? Which ones are possible?
  2. Unfortunately Gilbert had a marble floor at his party. If anyone spilled a drink, what volume of the floor was dissolved? (Assume that the glass was completely filled.)
  3. Bengt would like to have very long straws to the drinks. How long straws is it theoretically possible to have? (Assume that the drink has the same density as water, and that the drink is on Earth, and that the straws are pointing straight up.)
  4. Actually, the density of this particular drink is 568 kg/m^3. So how long can the straws be?
  5. The drink which Bengt has been planning consists of liquid ethene, which reacts with water in the stomach to form ethanol. What volume percentage of ethanol does this drink correspond to?

2009-01-17

Darts

Bengt and Sture are in the pub. They are drinking beer, obviously, and also playing darts. Bengt has invented a new kind of darts game. It is best played on a different kind of board, but in the pub they have to make do with the normal one.
The game goes like this: First Bengt throws three darts, hitting a certain combination of fields. Then Sture throws three darts, trying to copy Bengt's throw by hitting the same fields (in any order). If he succeeds, the round is over and no one gets any points. If he fails, Bengt gets to try to copy his own combination. If he succeeds in the first attempt, he gets three points. If he fails, he gets a second attempt; if that succeeds, he gets two points, otherwise he gets no points. Then the round is over. In the next round Sture starts by setting the combination, and then they alternate like that until they have played an even number of rounds and someone has achieved a previously agreed upon score.
(There is also a rule that you are not allowed to use the same combination twice, but that has no impact on this question.)

What makes this game different from other darts games is that you not only need to have good throwing skills, but you also need to be able to assess your throwing skills well. If you start by throwing a really easy combination, the opponent will almost certainly copy it in his only attempt, and if you start with a really difficult combination, you are very unlikely to be able to copy it even in two attempts. So the trick is to know the probability that you will be able to pull off a certain combination, and then to choose a combination that will have the right probability.

  1. Which probability would be the best to have in your first throw? Assume that both players have the same skill level (and that it remains constant, despite the beer).
    In other words: Provided you have already done a first throw (of three darts), and it is such that each of the following throws has a probability p to copy it, what would you want p to be?
  2. When you make your first throw, what probability should you aim for? Assume that if you miss the intended target with any of the darts on your first throw, you will not get any points (because you will hit something that is either too hard or too easy to copy).
  3. What if you make a slightly more complicated assumption - if you miss with a dart, that dart will hit something that is trivial to copy (such as the wall), but you can still hope to make it a challenging throw by hitting something with the other darts. Which probabilities should you aim for with each of the three darts? (It may depend on whether you hit with the previous darts, so there are seven cases in total.)

2009-01-10

The Paper Dragon

Bengt has a long strip of paper. He puts it on the table, and folds it in half. He lifts up the right hand end, folds it over and puts it on the left hand end. Then he repeats that same procedure several times. When he unfolds the paper, it has a funny curly pattern, which supposedly looks like a dragon. But Bengt is a true scientist and is not content to note that the pattern looks nice. He wants a formula.
If you call a fold in one direction 0 and in the other direction 1, the result is
0010011000110110001001110011011...
  1. Find a way to reproduce the sequence mathematically.
  2. Find a recursive formula that describes it, that is, the folding direction as a function of the direction of one or more previous folds.
  3. Find a non-recursive formula, that is, the folding direction as a function of only the number of fold.

2009-01-01

Fireworks

It's New Year's Eve, and Bengt is setting up the fireworks. Usually he's not much for safety regulations, but this day he has all of a sudden decided to be really paranoid about safety. He wants to make absolutely sure no one can be hit by a rocket.
He knows that the rocket burns for a time t, and that it is supposed to reach a height h. But that's assuming that it moves straight upward.
While it burns, the acceleration has a constant size, and after that the rocket is only affected by gravity.
Naturally, Bengt would like to know how far the rocket might reach if it is fired at an angle.

  1. Assuming that the rocket doesn't turn, and that there is no air friction, can we safely assume (for the purpose of these calculations) that the rocket has constant mass?
  2. Come to think of it - assuming that there is no air friction, can we safely assume (for the purpose of these calculations) that the rocket doesn't turn?
  3. Come to think of it - can we safely assume (for the purpose of these calculations) that there is no air friction? Remember we're trying to determine an upper bound for the distance, so we just want to know if there is any way the rocket could get a longer range due to air friction.
  4. Assuming all those things, how far could the rocket reach?
  5. Sture leeringly remarks that Bengt probably has no interest in safety, that he just wants to know how many of his neighbours' houses he could bomb. Naturally we trust Bengt, but if he was trying to reach as far as possible - would it help to attach several rockets to each other? If he uses a bunch of n rockets all attached to each other, how far will the bunch go?
  6. What if he could change the time the rocket burns? Assume that the rocket gets the same amount of work done on it by the burning. How long should it ideally burn, in order to fly as far as possible?
  7. And how far would that be?
  8. If the rocket burns for b times longer, still with the same amount of work, how far will it go?