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?