Red Light Optimisation

Bengt is cruising around in his car, when suddenly he comes to a red light. The annoys him very much. In order to minimise his annoyance, he would like to make sure he gets through as quickly as possible. The thing is, if he drives all the way up to the light and stops, it will take several seconds to accelerate, and we can't have that, now can we?

We can safely assume that Bengt is driving at the maximum allowed speed, v, when he is not taking the traffic light into account. As a simplification, we'll assume that his maximum acceleration (forward) is a constant m, and that he can break (i.e. accelerate backward) without limit. The question is of course, what speed should he drive at, as a function of something suitable, like perhaps the distance to the traffic light, or some kind of time?

1. First, let's simplify a little further, by assuming that Bengt just wants to maximise his speed at the moment the light turns green. In a sense, we are saying that m is close to zero, which is somewhat insulting to Bengts car, but might make the calculations easier.
   Suppose now that we have no idea when the light will turn green. Can the problem be solved? If so, please do.
2. Suppose instead that the light turns green after an average time T, and the probability that it will happen in the next moment remains constant - that is, the red light has a "half-life".
3. Suppose instead that Bengt sees the light turn red at a particular time (or distance, if you like) and that it will turn green at any time between then and another time, which, for simplicity, we might call "one minute later".
4. Now do a), b) and c) again, but remove that simplification about the acceleration being close to zero.
5. Pick the option that you think is the most realistic, and find how much time he would save by doing all this rather than driving as fast as possible up to the light.

A Simple Board Game

Bengt is in the mood for designing a game. But his imagination is not at its best. He was up late last night, and... well, never mind that. He's making a rather simple board game, one of those where you roll a die and walk along a path. As soon as you reach (or pass) the finish line, you win.
The game is played with a special k-sided die. Just to be completely clear, the die shows each integer from 1 to k inclusively, with equal probability. At the start of the game, you also get a few cards which do various things, but apart from that nothing much happens in the game. There's supposed to be several types of cards, but so far Bengt has only come up with one type. This card can be played right after you've rolled the die, and lets you double the number of steps. Thus, obviously, it should be played when you've rolled a rather high number. Bengt calls this card "the doubling card". He really is unimaginative today.

1. Imagine you are n steps from the finish line, and you want to reach it in as few turns as possible. You have one doubling card. You roll the die. What is the minimum value on the die roll for which it is a good idea to use the card?
   
2. What if you have c cards?
   
3. If there are other players involved, and your goal is not to win as soon as possible but to beat the other players, does that affect the optimal strategy at all? If so, how?

A Cup of Tea

Bengt is back from his winter holiday, and he's exhausted. So he sits down by the fireplace with a nice cup of tea. The cup is cylindrical, at least on the inside, with radius r and height h. He has three footstools; first he had two, because he figured he has two feet, but then he added another to put his cup of tea on. The stool-tops are horizontal - Bengt's woodworking skills are dubious, but he didn't make these. As the tea stands on the stool, the depth of the tea is d. Then he picks up the cup, and tilts it towards him, as people tend to do when drinking.

1. At what angle is the bottom no longer covered by tea?
2. At what angle does the tea reach the top of the mug, and thus Bengt's mouth?
3. Come to think of it, Bengt is curious about how fire works. It emits light, that's plain to see. But is that light just black body radiation, or is it some sort of chemical process that emits light in itself?