Christmas Equation

It's Christmas Eve, and Bengt is celebrating by dancing around a new equation. It looks like this:
√(x²+y²)=5-z+f(z)-√(5z)
where f the floor function, that is, f(z) is z rounded down to the nearest integer.

What is this the equation for?

Running Aliens

Above Bengt's house floats a spaceship. It's big and round - basically what you would call a "flying saucer". It is held up by a force which exactly matches that of gravity, so that the spaceship stays hovering at the same height.
In the ship is a little green alien, who likes to get his exercise in the morning by running around the perimeter of the spaceship. Around and around in a circle, at an amazing speed. To avoid the spaceship starting to turn the other way, he has an identical friend alien who runs the other way. The other alien thinks this is a bit of a drag, but he still has the same physical properties.
The aliens are running so fast that you have to use relativistic equations to calculate their momentum. There are two ways of doing that; either it's m * v / gamma, or you use speed-dependent mass, which makes it m * v. Either way, the alien can never reach the speed of light, which is all good and well.
But if you use speed-dependent mass, the alien will get heavier. That means gravity on the spaceship as a whole increases, and it falls down. On the other hand, if you don't use speed-dependent mass, he doesn't get heavier, so it doesn't fall down.

Which is it? Does the spaceship fall down, or not?

A Big Cube and a Little Cube

On his desk, Bengt has two cubes. The small cube has the side 2 dm, and the big cube has the side 1 dm. The reason why the 1 dm cube is "big" is that Bengt claims it has more than three dimensions. Now Bengt wants to place the small cube inside the little cube. (The thickness of the walls of the cubes are negligible, so that for example if you had two identical cubes you could place one inside the other.)

1. How many dimensions must the big cube have for the little cube to fit inside it?
   
2. What if they were spheres instead of cubes?
   

A Slowly Growing Number Sequence

Bengt has found a number sequence. It starts like this:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12, 4, 6, 2, 10, 2, 6, 10, 2, 10, 2, 6, 18, 4, 2, 4, 6, 6, 8, 6, 6, 22, 2, 10, 8, 10, 6, 6, 8, 12, 4, 6, 6, 2, 6, 12, 10, 18, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 34, 6, 6, 8, 18, 10, 14, 4, 2, 4, 6, 8, 4, 2, 6, 12, 10, 2, 4, 2, 4, 6, 12, 12, 8, 12, 6, 4, 6, 8, 4, 8, 4, 14, 4, 6, 2, 4, 6, 2, 6, 10, 20, 6, 4, 2, 24, 4, 2, 10, 12, 2, 10, 8, 6, 6, 6, 18, 6, 4, 2, 12, 10, 12, 8, 16, 14, 6, 4, 2, 4, 2, 10, 12, 6, 6, 18, 2, 16, 2, 22, 6, 8, 6, 4, 2, 4, 8, 6, 10, 2, 10, 14, 10, 6, 12, 2, 4, 2, 10, 12, 2, 16, 2, 6, 4, 2, 10, 8, 18, 24, 4, 6, 8, 16, 2, 4, 8, 16, 2, 4, 8, 6, 6, 4, 12, 2, 22, 6, 2, 6, 4, 6, 14, 6, 4, 2, 6, 4, 6, 12, 6, 6, 14, 4, 6, 12, 8, 6, 4, 26, 18, 10, 8, 4, 6, 2, 6, 22, 12, 2, 16, 8, 4, 12, 14, 10, 2, 4, 8, 6, 6, 4, 2, 4, 6, 8, 4, 2, 6, 10, 2, 10, 8, 4, 14, 10, 12, 2, 6, 4, 2, 16, 14, 4, 6, 8, 6, 4, 18, 8, 10, 6, 6, 8, 10, 12, 14, 4, 6, 6, 2, 28, 2, 10, 8, 4, 14, 4, 8, 12, 6, 12, 4, 6, 20, 10, 2, 16, 26, 4, 2, 12, 6, 4, 12, 6, 8, 4, 8, 22, 2, 4, 2, 12, 28, 2, 6, 6, 6, 4, 6, 2, 12, 4, 12, 2, 10, 2, 16, 2, 16, 6, 20, 16, 8, 4, 2, 4, 2, 22, 8, 12, 6, 10, 2, 4, 6, 2, 6, 10, 2, 12, 10, 2, 10, 14, 6, 4, 6, 8, 6, 6, 16, 12, 2, 4, 14, 6, 4, 8, 10, 8, 6, 6, 22, 6, 2, 10, 14, 4, 6, 18, 2, 10, 14, 4, 2, 10, 14, 4, 8, 18, 4, 6, 2, 4, 6, 2, 12, 4, 20, 22, 12, 2, 4, 6, 6, 2, 6, 22, 2, 6, 16, 6, 12, 2, 6, 12, 16, 2, 4, 6, 14, 4, 2, 18, 24, 10, 6, 2, 10, 2, 10, 2, 10, 6, 2, 10, 2, 10, 6, 8, 30, 10, 2, 10, 8, 6, 10, 18, 6, 12, 12, 2, 18, 6, 4, 6, 6, 18, 2, 10, 14, 6, 4, 2, 4, 24, 2, 12, 6, 16, 8, 6, 6, 18, 16, 2, 4, 6, 2, 6, 6, 10, 6, 12, 12, 18, 2, 6, 4, 18, 8, 24, 4, 2, 4, 6, 2, 12, 4, 14, 30, 10, 6, 12, 14, 6, 10, 12, 2, 4, 6, 8, 6, 10, 2, 4, 14, 6, 6, 4, 6, 2, 10, 2, 16, 12, 8, 18, 4, 6, 12, 2, 6, 6, 6, 28, 6, 14, 4, 8, 10, 8, 12, 18, 4, 2, 4, 24, 12, 6, 2, 16, 6, 6, 14, 10, 14, 4, 30, 6, 6, 6, 8, 6, 4, 2, 12, 6, 4, 2, 6, 22, 6, 2, 4, 18, 2, 4, 12, 2, 6, 4, 26, 6, 6, 4, 8, 10, 32, 16, 2, 6, 4, 2, 4, 2, 10, 14, 6, 4, 8, 10, 6, 20, 4, 2, 6, 30, 4, 8, 10, 6, 6, 8, 6, 12, 4, 6, 2, 6, 4, 6, 2, 10, 2, 16, 6, 20, 4, 12, 14, 28, 6, 20, 4, 18, 8, 6, 4, 6, 14, 6, 6, 10, 2, 10, 12, 8, 10, 2, 10, 8, 12, 10, 24, 2, 4, 8, 6, 4, 8, 18, 10, 6, 6, 2, 6, 10, 12, 2, 10, 6, 6, 6, 8, 6, 10, 6, 2, 6, 6, 6, 10, 8, 24, 6, 22, 2, 18, 4, 8, 10, 30, 8, 18, 4, 2, 10, 6, 2, 6, 4, 18, 8, 12, 18, 16, 6, 2, 12, 6, 10, 2, 10, 2, 6, 10, 14, 4, 24, 2, 16, 2, 10, 2, 10, 20, 4, 2, 4, 8, 16, 6, 6, 2, 12, 16, 8, 4, 6, 30, 2, 10, 2, 6, 4, 6, 6, 8, 6, 4, 12, 6, 8, 12, 4, 14, 12, 10, 24, 6, 12, 6, 2, 22, 8, 18, 10, 6, 14, 4, 2, 6, 10, 8, 6, 4, 6, 30, 14, 10, 2, 12, 10, 2, 16, 2, 18, 24, 18, 6, 16, 18, 6, 2, 18, 4, 6, 2, 10, 8, 10, 6, 6, 8, 4, 6, 2, 10, 2, 12, 4, 6, 6, 2, 12, 4, 14, 18, 4, 6, 20, 4, 8, 6, 4, 8, 4, 14, 6, 4, 14, 12, 4, 2, 30, 4, 24, 6, 6, 12, 12, 14, 6, 4, 2, 4, 18, 6, 12, 8...

1. What is this number sequence?
   
2. Will it contain arbitrarily large numbers? Prove it.
   
3. Of the 1000 numbers shown here, the biggest number is 34, and it appears quite early. Is there any particular reason for that?
   
4. Why does the sequence 6, 6, 6 appear in so many places? Does it have something to do with the devil?
   
5. Will any other number ever appear three times in a row? Prove it.
   
6. If you have plenty of spare time: Find the next number in the sequence.
   
7. (hard) Near the beginning, there are a lot of twos. Will they eventually run out, or is there an infinite number of them?

Technology Logo

Bengt's friend Alban has started studying at LTH, the Faculty of Engineering at Lund University. He therefore has a sweater with the logo of the student union, TLTH (http://www.tlth.lth.se/). It consists of a square inside an equilateral triangle inside a circle.
Bengt is thinking about whether this is the optimal way to pack these shapes.

1. If the circle has the area 1, what area does the square have? (It is as big as possible.)
   
2. If the shapes contain each other in a different order, one might get a different ratio between the outermost and the innermost. If the outermost shape always has area 1, in which order should they be placed to make the innermost as big as possible, and how big will it be?
   
3. What if they are in three dimensions? A sphere, a regular tetrahedron, and a cube, with the outermost having volume 1 - how big will the innermost be?
   
4. While you're at it: n dimensions?

A Different Kind of Relativity

Bengt is sitting in his hammock, thinking. The hammock is doing a pendulum motion, but that has nothing to do with this problem.
"In classical physics," Bengt thinks, "momentum is always mass times velocity, m * v. In relativistic physics, momentum is usually taken to be m * v / gamma. But if you really want, you can decide that it is m * v after all, and use speed-dependent mass. And since the earth is moving, it gets a higher mass."

1. Assuming that the earth is spinning, but that its centre of mass is at rest, how much does its relativistic mass increase compared to if it was standing still?
   
2. If the earth is also spinning around the sun, how much does that increase its mass?
   
3. Bengt's (rest) mass is 100 kg. If the mass of the earth increases, gravity should get stronger. Also, Bengt moves along with the earth, so his mass should increase too. If Bengt is at the equator, and the earth moves as in b), how much weight does he gain?
   
4. The rotation of the earth also leads to a centrifugal force, so that Bengt seems to get lighter instead. If Bengt is at the equator, how much is his apparent weight loss?
   
5. The rotation of the earth has some other consequences. Normally, the circumference of a circle (or a sphere, in this case) is pi * D, where D is the diameter. But if you try to measure the equator, it is because of length contraction not quite pi * D. But again, if you really want, you can decide that it is pi * D after all, and use speed-dependent pi. In that case, how much does pi change when you measure the equator?
   

Equation with Three Unknowns and One Well Known

Bengt is sitting in the town square eating ice cream. He looks like he is deep in thought. Sture walks by and comes up to him.
"Hi there, Bengt! What are you thinking?" asks Sture.
"Oh, hi. Well, I was just thinking, if x^2 + y^2 + z^2 + 1 = x + (2y + √(23)z) / 3, what is then x, y and z / √(23)? All the numbers are real."
"But that's impossible to solve, isn't it? It's three unknowns but only one equation."
"I'm sure you can do it. You're good with numbers."

Is it possible to solve? What is the answer?

A Different Kind of Boat

Bengt has a thick metal sheet. It is in the shape of a hexagon with the side 1 m. Now he wants to make it float. Here's his idea:
He will drill a lot of holes in the sheet. Then he will put a very thin metal sheet on each side, so that the air can't escape.
He has drill bits in all sizes. The holes can be infinitely close, but must not overlap. The volume of the thin sheets can be neglected.

1. If he chooses the best possible size of the bit (but only one size), which fraction of the mass of the sheet will remain?
   
2. What if he chooses the worst possible?
   
3. What if he uses the best possible combination of two sizes?
   
4. What if he uses the best possible combination of n sizes?
   
5. How many sizes would he need to make the plan work (get the boat to float on water) if the sheet is made of iron? How about lead? Aluminium?
   

The Great Clock

Bengt has a big clock on his wall. It's an old mechanical clock, powered by a spring. The spring keeps a pendulum wheel in motion. The pendulum wheel is spinning back and forth. The torque on the wheel from the spring is constant in magnitude but with alternating direction.
The wheel, which is made of copper, 1 mm thick, and with 1 dm circumference, moves a quarter of a turn from its equilibrium position in each direction. Each time it reaches one side (but not when it reaches the other) a big gear moves one step. On the same axis as this gear is small gear, which meshes with an identical big gear, moving it ahead one step for every turn of the small gear. This second big gear moves the second hand on the clock, moving with the same angular velocity as it. On the same axis as this second big gear is another small gear, which similarly meshes with a gear for the minute hand. And so on.
Apart from the normal hands, this clock also has a hand which completes a turn in one year. It is one meter long.
"Suppose that the gears and the hands have no mass." thinks Bengt. "And suppose that the clock won't break if I hang in it." So he does.
Bengt jumps up and grabs a hold of the tip of the year hand. It is autumn, so it is pointing to the left, moving upwards.

1. How big is the torque on the pendulum wheel from the spring?
   
2. Even though Bengt hangs in the year hand, the clock doesn't stop. How heavy would he have to be to stop the clock?
   
3. Bengt's mass is 100 kg. How much behind would the clock be if he hangs there for 24 hours?
   
4. "When spring comes," thinks Bengt, "I'm going to hang in the year hand again, to see if I can make it go twice as fast." How heavy would Bengt have to be to succeed in this?
   

Espresso Numbers

"The real numbers are not a countable set." says Alban.
"Oh really?" says Bengt, in a somewhat disdainful tone.
"Really."
"So, if I give you a countable set, you can tell me a real number which is not in my set?"
"Yes... I suppose."
"Bet you a beer you can't."
"You're on."
"Okay, listen to this: There is a set S, which contains all the symbols valid for expressing a number. Digits, decimal points, fraction signs, and so on. You can throw in some arithmetic symbols too if you like. The point is, it's a finite number of symbols."
"Uh-huh."
"With those, you can put together expressions of finite length. There is clearly a countable set of such expressions. Now, we define the set U, which is the set of all numbers that can be defined using such expressions. We can call them the expressible numbers."
"Or we can call them the espresso numbers. Because I like espresso."
"No. First, that doesn't sound mathematical. Second, I don't like espresso."
"Oh."
"So, there you go. Find me a real number that is not expressible."
Alban thinks for a while.
"Umm... pi?"
"Well... I could say that the pi symbol is in S, but that would be no fun. Or I could bring up Lebniz' and Gregory's formula. But that's not necessary. Because if pi is not in S, you have to use some other way to express the number you mean. Can you explain what pi is?"
"Hmm. I guess that should be possible somehow... but we know it's irrational, so it has an infinite non-repeating sequence of digits."
"But so does the square root of two. And that's obviously expressible."
"Damn you and your strange problems."
"Yeah yeah. Sore loser."

Is there any such number? Since it's well known that the set of real numbers is not countable, there must be some flaw in Bengt's argument - what is it?

The Hotel Manager

Bengt is thinking about getting a job. He thinks it would be cool to be a hotel manager. This is obviously never going to happen, because Bengt is far too lazy to be a hotel manager, possibly too lazy for any job at all, but he still likes to think about it. But he can't decide what would be the optimal ratio between single and double rooms in the hotel.
Whenever guests arrive at the hotel, there is a probability p that it is a couple, and a probability 1-p that it is a single person. A single person can occupy any room, either a single or a double. A couple can occupy one double room, or (although it might make them somewhat dismayed) they can occupy two single rooms. A double room is d times bigger than a single room, and there is only a certain amount of space in which to build rooms.

To avoid as far as possible having to turn guests down, which fraction of the rooms should be double rooms?

A Taste of Combinatorics

It is summer, and Bengt is on the beach. He is hungry, so he goes to buy ice cream.
"How many flavours are there?" he asks the salesman.
"There are n flavours today."
"Then I would like a number of cones, each with k different flavours."
"And no two cones must be identical? Ah, that's a classic problem of combinatorics."
"No, that's too easy. And that way, I would get some cones that are almost identical. No, I want there to be no two cones that have p flavours in common."

1. Let's start with something easy: Suppose p = 1; in other words, no two cones have any flavours in common. How many cones will Bengt get?
   
   
2. Then the classic problem, where p = k; no two cones are identical. How many cones does Bengt get?
   
   
3. Finally, the problem Bengt is really concerned with, for a general p. How many cones?

The Alumium Can

"Consider this can." says Bengt.
"Okay." says Sture.
"It is a can of beer."
"Ah. One of the great inventions of mankind."
"Indeed. However, I'm thinking of returning this particular can to the store. It's flawed."
"How?"
"Well, we can assume that the outside shape of the can is a cylinder."
"Indeed."
"The space inside it is also a cylinder. The can itself is thus a cylinder minus another cylinder."
"Indeed."
"So the top and bottom are also cylinders, and they have the same thickness. The side wall has the same thickness all around, but not the same as the top and bottom. You'd expect all the walls to be very thin, but they're not; it's a very sturdy can."
"Right."
"Inside the can is beer and air. The beer has roughly the same density as water, but the difference should not be neglected. The density of the air, however, can be neglected. The can itself is made of pure alumium."
"You mean aluminium? Or aluminum, as the Americans say?"
"Well yes, but it should properly be called alumium."
"I see. So what's the problem with it?"
"It's not full. I don't know what fraction of it is full, and I have to find out without opening the can, because otherwise I obviously can't return it."
"Oh. Got yourself a good problem there, Bengt."
"Indeed."

Let the thickness of the top and bottom walls be b, the thickness of the sides be s, the density of the beer d, and the fraction of the inside which is filled with beer f.

1. If you know b, s and f, you can find d by weighing the can. Obviously you can also measure the outside size of the can. What is the formula for d?
   
   
2. If you know d, how can you find b, s and f? You must not open the can, or deform it. Bengt also does not have an x-ray camera. It's a mechanics problem, folks!
   
   
3. If you don't know either of those variables, how would you go about solving it? Is it still possible?

Two Very Long Steel Wires

In his garage, Bengt has two long steel wires, wound up into balls. They are very, very long indeed.
One of them is so long it could go all the way around the world. The other one is 1 m longer. They are 1 mm thick.
(Assume that the Earth is spherical, and that there is no air in the balls.)

1. Bengt imagines laying out the two threads along the equator. One of them is touching the ground, the other one is a small distance away from it (same distance everywhere).
   How far from the ground is the second wire?
   
   
2. Bengt imagines pulling up on the long wire, so it is taut against the ground on the other side. How far from the ground can he pull the thread?
   
   
3. What fraction of its length would be touching the ground?
   
   
4. Bengt imagines laying out the threads (like in question a) and then moving the shorter thread to the north, until it is the same distance from the ground as the other one. How far does he move it?
   
   
5. What is the diameter of the balls?
   
   
6. The balls are lying next to each other. What gravity force do they exert on each other?
   
   
7. The floor of the garage has zero friction. Unfortunately it is not entirely horizontal, so Bengt has had to tie the balls to the wall. If he lets one of them loose, and it is held in place by the gravity from the other, what is the maximum possible angle of the floor?

Pacman's Adventures in Curved Space

Bengt and his friend Alban are playing Pacman on Bengt's new (and very expensive) calculator. The screen is 1x1 dm. Alban is, unlike Bengt, younger than Pacman, and he has played a lot of computer games. Neither of them knows much about the theory of relativity, but Bengt has a habit of discussing things that he doesn't understand.
"This is a bit like the theory of relativity." says Bengt.
"Really? How?" says Alban.
"Well, when Pacman leaves through the hole in the wall in the middle of the top edge of the screen, he immediately reappears at the corresponding hole at the bottom, and the other way around."
"You mean that he has moved faster than light?"
"No, that's not what I mean. Don't try to tell me what I mean. No one but me knows what I mean."
"True... so, what do you mean?"
"That he comes back, it's like if space is curved. As if he lived on a cylinder. The Y coordinate repeats itself; it's a cyclic coordinate."
"Uh... that's not what 'cyclic coordinate' means. It means a coordinate which the lagrangian is independent of, or something like that."
"I don't care. If a coordinate repeats itself cyclically, I will call it a cyclic coordinate. And apart from cyclic coordinates, there are also limited coordinates, which end, and adifatic coordinates, which neither end nor repeat themselves. On a cylinder, one coordinate is cyclic, and the other is limited."
"Okay, if you say so..."
"If there were holes on the left and right edges as well, so he could go around in the X coordinate as well, Pacman might get the impression that the world was round. Spherical. Or that he was living on on a doughnut. One might also imagine parallel universes, in the shape of doughnuts linked as a chain. I call the theory "the catananese doughnut theory".
"Blast! Now I died again. Could you please stop talking geometry? I'm trying to concentrate."
"No I won't. It is also possible to let him leave and reappear not only in the middle, but anywhere on the top or bottom edge, and we can imagine that the screen is infinite in the X dimension, so that that coordinate is adifatic. Then it would be like an infinitely long cylinder, and if Pacman seems to move on a straight line on the screen, he is in fact moving in a spiral on the cylinder surface, and depending on his direction the time it takes to get back to the same Y position varies. It's just like in string theory."
"Sure, sure, a spiral, on a doughnut..."
"No, that wouldn't work. One can also imagine that he can leave anywhere on any edge, but it will become neither a sphere nor a doughnut."
"Cut it out, I'm getting a headache."
"Okay. Beer?"
"Yes please."

1. If Pacman lives on a finite cylinder, what is the surface area and volume?
   
   
2. What if it is a sphere? What is the surface area and volume then?
   
   
3. Since the screen is a square, is it really possible that he lives on a doughnut? How big is the hole?
   
   
4. If Pacman is spiralling along an infinite cylinder, what is the time until he returns to the same Y coordinate, as a function of the angle between his movement and the X axis?
   
   
5. If he can leave anywhere on any edge, under what conditions will he ever come back to the same point (without turning)? How far will he then have moved?
   
   
6. If he can leave anywhere on any edge, why is it impossible that he is on a sphere or a doughnut?
   
   
7. In that case, if he is moving on the surface of some shape, what kind of shape is it? How many dimensions does it have?
   
   
8. What is it's volume? (That is, the equivalent of volume in however many dimensions it has.)
   

Rapunzel

Bengt is a big fan of fairy tales. Unfortunately, he is the kind of person who always ruins them by pointing out things that are not logical.
One example is the story of Rapunzel. She was a girl whose hair was very long and beautiful, and grew very fast. Every time she cut her hair, it grew twice as fast.
She was imprisoned by a witch, who put her in a tower. The tower was as tall as twelve men. It had no stairs and no door, so the only way to get to Rapunzel was to call for her to let down her hair through the window, and then use the hair to climb up. Eventually a prince comes along, and uses this method to enter the tower.
There are many things about this that makes Bengt think.
"Is it possible to climb up someone's hair like that? It would be pretty uncomfortable for the girl... but I guess she could just wrap it a couple of times around something at the end near her head, before letting it out."
He assumes that all men, like himself, weigh about 100 kg, and are about 180 cm tall. He also assumes that Rapunzel is only 120 cm, and weighs 40 kg. This is clearly a rather short girl, but Bengt happens to like short girls.
Bengt pulls out one of his own hairs, to test its strength. It turns out it can carry the weight of ten grams, and can be extended by about 5% of its normal length.
Next, he wants to see how thick his hair is. He gets out a red laser pointer, and points it at the single hair. On the wall two meters away, he sees the diffraction pattern. The first black band is 15 mm from the center.
Bengt decides Rapunzel was probably blonde. His own hair is some sort of brownish colour. He happens to know that blonde people have about 50% more hairs, and they are about half as thick. On the other hand, they both have rather straight hair, which means that the cross section of the hair is circular.
Bengt realises he needs to do one more test. He immerses his hair in water, and somehow manages to determine that the volume of the displaced water is 0.8 ml. He also decides that "displace" is a funny word, and makes a mental note to start using it instead of "misplace", if only to annoy people.
He dries his hair, happy that it is only a decimeter long, so no one will be climbing in it any time soon.

1. If Bengt was to cut his hair only when it reached the floor, and then cut off all of it, he would have to cut his hair once every 15 years. If Rapunzel, with her ever-increasing hair growth speed, did the same, what would eventually happen?
   
   
2. How long could she keep doing that? (Assume that she starts off with the same growth speed as Bengt.)
   
   
3. Suppose instead that her hair grows at normal speed. How long would it take for the hair to reach the ground outside the tower?
   
   
4. Sometimes the prince might need to make a hasty exit. He can then grab a hold of the hair, and jump out through the window. Assuming that he can hold on to the hair, for how tall towers would this be possible?
   
   
5. Without any jumping, how many princes could hang on Rapunzel's hair at once?
   
   
6. For best result, should they be spaced equally, or all hanging at the end?
   

Solid Geometry and Booze

Bengt is lying in his pool, thinking. The pool is 25 m long, 16 m wide and 2.5 m deep without Bengt.
"Suppose that my weight will remain constant until I die, and that I will live a hundred years." thinks Bengt. Considering his food habits, those are somewhat unrealistic assumptions, but he doesn't care.
"Also suppose that my birth weight was 3 kg, and my weight so far has increased linearly. 10% of my body weight is blood, and the same is true for all humans."
Here, Bengt's thoughts are interrupted by the sound of the doorbell. Bengt leaves the pool. The water level sinks 0.25 mm.
It's Bengt's old friend Sture who comes to visit.
"Hello Sture." says Bengt. "Come on in. Would you like a drink?"
"Yes please. Some ethanol would be nice."
Bengt pours a glass for each of them.
"What are you up to these days?" asks Bengt.
"Oh, nothing much. I've started studying calligraphy, to write more beautifully, and stenography, to write faster. I'm hoping they will negate each other, so I can get a degree without learning anything."
"Why not economics and moral philosophy?"
"I'm not lazy enough. So, how about you?"
"I'm waiting for my evil twin to come back to Earth. He was sent out into space when we were born. He says he's going to come back this Friday, when I turn 50, and celebrate that he's only turning 49."
"Oh, wow. I had no idea you were so incredibly old. So what is this spaceship he has?"
"Well, it has constant acceleration up to a point, then constant acceleration in the opposite direction to turn, and then in the first direction again to stop, so that it has velocity zero relative to me when it lands over there." says Bengt, pointing. "I, on the other hand, have basically not moved since he left."
"But don't you exercise?"
"Yes, but it's negligible."
"So what fuel does the spaceship use?"
"Ethanol, just like you."
"It's a very good fuel. Have I told you about when I climbed Mount Everest from sea level driven only by booze?"
"Yes, many times. But it can't have been so difficult, because you only weighed 50 kg back then."
Sture tells the story again, just because it's so good. Then he leaves, and Bengt gets ready to jump back into the pool.
"Imagine if this pool was filled with gold rather than water." he thinks. "Then it would weigh as much as my brother's space ship. But it would be difficult to dive." thinks Bengt, and dives.

1. How far away has the spaceship been, from Bengt's perspective?
   
   
2. The space ship needs to carry its own weight, Bengt's brother (who weighs as much as Bengt) and the fuel. It has 100% efficiency, and ejects the exhausts.
   If it is powered by combustion, how much fuel does it need?
   (Remember that it also needs oxygen. The oxygen for Bengt's brother is negligible.)
   
   
3. If instead it is powered by fusion of the hydrogen atoms in the ethanol, how much fuel does it need?
   
   
4. If Sture climbed to the top of Mount Everest from sea level, without wasting any energy on anything other than the vertical movement, what was his blood alcohol content when he set out? (Assume that the ethanol is added to the blood mass and the body mass.)
   
   
5. Bengt likes to stretch out in the pool. If he could stretch out really really much, so he covered most of the area of the pool, would the surface tension be enough to carry him?
   
   
6. If the speed of light in vacuum is 1, how many m^4 is Bengt's whole life?
   

The Hourglasses

Bengt is visiting his grandmother, who lives in a very small and very old-fashioned village far out in the countryside. The village is so old-fashioned that there are no mechanical or electric clocks, only hourglasses.
In every house is an hourglass. Every whole hour the sand runs out in all of them, and the villagers instantly turn them.
But Bengt's grandmother is very very old, and now she has forgot to turn it. Bengt has to fix this by going to one of the neighbours to look at their hourglass. The problem is that one can only read the time when the hourglass is being turned, and the hourglasses are mounted on walls so you can't carry them away.
Bengt can walk with perfectly constant speed, and it takes only a few minutes to get to the nearest neighbour. He doesn't know exactly how fast he walks or how far it is, but he does know that there is a well at exactly half the distance.

How should Bengt go about making granny's hourglass catch up with the rest of the village? (Shouting is not allowed.)

A Wonderful Dream

Bengt is asleep in his bed, dreaming. In the dream he is in free fall with no air friction. It is almost dark; the only things visible are Bengt's glow-in-the dark watch, which shows that he is not supposed to wake up for another six hours, and a brightly glowing point of reference far away.
"What a lovely dream!" thinks Bengt. "An endless, homogeneous gravity field. The perfect opportunity to perform a really incredible twist... jump, or something. By moving my arms, I can make myself spin around and around."
Suppose Bengt's body is a cylinder, with height 1.8 m, radius 0.25 m, and mass 90 kg. His shoulders are represented by massless spheres, with radius 0.05 m, attached to the body on opposite sides. His arms are also cylinders, with length 0.8 m, radius 0.05 m and mass 5 kg each. The base of the arm-cylinder intersects the shoulder-sphere, dividing it in half, and spins around it.
He lets his left arm point forward and his right arm straight out to the side, then he moves them 90 degrees to the left, and then he brings them down parallel to the body. This he can repeat once per second.

1. If Bengt's mass is constant, what angular speed can he reach before waking up? (Assume that he can take any angular speed without expanding or exploding. It is a dream, after all.)
   
   
2. If the gravity is the same as on Earth, how far will he fall?
   
   
3. If all the energy needed for the rotation is taken from fat in his body, how much lighter will he get? (Assume that the fat is homogeneously distributed in Bengt.)
   
   
4. If he finally lands in his bed without bouncing back up, what change in velocity will the Earth get?