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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?