## Difficult Physics and Game Theory Problems.

Some are straightforward, some are trick questions, none are lengthy I know the answers to some but not all.

If you have a favourite difficult physics problem, I'd be delighted to add it to the collection

Statics 1/ A ladder mass m rests at angle theta against a rough wall on rough ground each with coefficient of friction mu. A man (point mass) climbs the ladder. What is the maximum weight which the man can be before the ladder slips?

Statics 2/ A cowboy ties his horse to a pole. Rather than making a knot, he relies on friction: does it matter how many times he loops the rope around the pole?

Dynamics 1/ A rope hangs over a pulley, on one side is a monkey of mass \$m\$ on the other a large bunch of bananas, also \$m\$. The monkey climbs the rope - what happens to the bananas?

2/ Cars leave a city along roads leaving sufficient gap between them that they can stop in that distance using constant acceleration of -g. When the speed limit is 50mph, it takes one hour to get all the cars out of the city: what speed limit would enable the city to be cleared in 30 mins?

3/ A boy standing 10m from a straight railway track throws a stone at 10m/s horizontally to hit the nameplate on the side of an engine passing at 50m/s. What is the last moment at which he can throw it? (ignore gravity, air resistance etc)

4/ Show that the Law of Conservation of Linear Momentum is a special case of Law of Conservation of Angular Momentum.

Fluids A train passes a station at 50m/s with its window open is the pressure higher inside or outside? (Bernoulli equation, P + dgh + dv*v/2 is a constant, d being density. Consider observers both on the train and the platform)

Blowing between two sheets of paper makes them move together (try it!): what happens if they are moving at the same speed as the blown air.

Gauss Law A closed surface contains zero charge inside, hence the area integral of the electric field is zero. The surface is expanded to contain 99% of the size of the universe, admitting only equally balanced amounts of positive and negative charge. What is meant by "inside"? If the other 1% which contains the entire net charge of the universe is regarded as "inside", does this mean that a closed universe has no net charge?

Small positively charged balls are at equilibrium in a circular dish. How many are needed around the edge before one goes to the middle?

Small positively charged balls are at equilibrium in a sphere at zero gravity. How many are needed around the edge before one goes to the middle?

Flies on the wall A rectangular room has dimensions 8 x 8 x 24 feet . Four imperial spiders, Godfrey, Stuart, Paul, and Lee, stand at the corners of the square wall. Each crawls directly towards the next (a moving target) with velocity 0.1ft/sec. How far do they each crawl before meeting in the middle? And how many times do they walk directly upwards.

A fly one foot above the bottom centre of one square wall decides to crawl to a point one foot below top center of the opposite square wall. What is the shortest crawling distance between these two points? (no, it isn't 32ft)

Experimental Physics A Lorentzian peak has amplitude A and full width at half height b. What is its standard deviation?

Galilean Relativity 1/ A 1000kg car accelerates from rest to 10 m/s, gaining 50kJ of energy. Viewed from the reference frame of a bus passing in the opposite direction the car accelerates from 10m/s to 20m/s, gaining 150kJ of energy. How much energy has the engine produced each case?

Special Relativity 1/ A speedboat approaches light speed. The captain sees the water moving by contract and become more massive, so he floats higher in the denser water. The harbourmaster sees the boat contract and become more massive, so it sinks. Do the crew get wet?

Waves Four wave packets of equal amplitude A and potential energy ~kA*A move in a 2D dispersionless medium as shown. When they reach the central point they superpose to form a wave of amplitude 4A. Is the potential energy of this wave ~16kA*A? If so, where did the energy come from, if not what is the potential energy?

Wavefunction collapse. An atomic excitation has finite width. Photons with energy just below the excitation are shone on the atom, sometimes exciting it to the excited state. When it decays, it gives off photons at its mean frequency, i.e. higher than the original. Where does the energy come from?.

Efficient Measurement Twelve gold balls are identical, except that one is gold plated lead or iron (and so may be heavier or lighter). Using a set of scales, how can the identity and composition of the plated ball be determined in three measurements?

Unnatural Selection Three men are fighting a duel (perhaps over misuse of the langauge?). They will fire shots alternately, one shot at a time, using gun A, B or C. Gun A hits the target 90% of the time, gun B 70%, gun C 50%. Which is the best choice of gun, and is it best to fire first, second or last?

Logic One Monday, a prisoner is told "You will be hanged on one day this week, but you won't know which day until it happens". And also: "If that statement is untrue, you will be pardoned"
He realises that if he gets to Sunday, then he'll know Sunday is the chosen day. So it can't be Sunday: he will be hanged in the first six days.
Then he realises that if he gets to Saturday (the last of the six days) then he'll know Saturday is the chosen day. So it can't be Saturday: he will be hanged in the first five days.
The same argument convinces him that there is no day on which he can be hanged without knowing it. The first statement is logically untrue, so a pardon is his.
To his total surprise, they hang him on Wednesday- what went wrong?.

Two state message passing There are ten prisoners in solitary confinement, each morning one is chosen at random to spend the day in an empty room with only a lightbulb for company - at the end of the day he can leave it on, or off. Once all the prisoners have visited the room, they will be released, provided someone asks. If anyone asks for release before everyone has been in the empty room, all the prisoners will be killed. Before the solitary begins, the prisoners try to form a plan to guarantee their safe release. Is there one? Is there a better one?

Gambling In a coin throwing game, you keep tossing a coin until you get a tail, at which point the game ends.
If you win a pound for every head thrown before you throw the first tail, how much do you expect to win?
If you win nothing for the first four heads (and nothing if a tail comes up), one for the fifth consecutive head, and twice as much for each head after that (e.g four pounds for seven consecutive heads), now how much do you expect to win?

Pirates Five pirates have collected a pile of gold coins. They go to sleep. Not trusting the others, the pirate on first watch divides the treasure into five equal piles. One coin is left over, which he tosses into the stream. He then hides his share. Later, the second pirate does the same, dividing the remaining coins into 5, finding one remaining and tossing into the stream, and hiding his share. Similarly, the third, fourth and fifth pirate. In the morning the five gather by the much-depleted pile, divide it into five, finding one remainder. How many coins were there originally?

Cube A cube of is made from 12 1ohm resistors. What is the resistance between opposite corners?

Pythagoras Prove that for any right angled triangle with integer length sides, one side is a multiple of 5.

Who knows? A simple card game involves picking at random one of eleven cards. The cards are numbered from 0 to 10. Highest card wins. Art and Bill each pick a different card, and don't show it to the other player. They correctly state-
Art "I don't know who won"
Bill "I don't know who won"
Art "I don't know who won"
Bill "I don't know who won"
Art "I don't know who won"
Bill "I won!"
Which cards did they pick?

Not magic Graham takes five cards at random from a 52-card pack. He selects one, and sorts and passes you the remaining four. From the order of these four cards you deduce the selected card. How? (no upside down cards or extraneous signals allowed!)