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?
   

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.)
   

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.
   

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?