PHYSICAL AND MATHEMATICAL PROBLEMS AND JOKES

1. You are drinking a can of beer in a shaky train. You don't want to empty the whole can immediately, but instead want to leave some of the beer inside, so that when you place the can on the (shaky) floor, its probability to fall aside will be the smallest. What is the amount of beer you should leave in the can? Assume that you know the density of beer and the dimensions and mass of the empty can.

2. Consider a cube of some material with a given density, floating in water (obviously the density of the material is less than that of water). Why is it that a cube of ice and a cube of very light plastic both float with the cube sides oriented horizontally (vertically resp.) whereas a cube of wood is flotaing with the sides oriented neither horizontally nor vertically? Find the critical densities for which something happens!

3. A drop of water is falling dwon to earth with acceleration g (neglect friction effects). Suddenly the drop enters a misty cloud (which is not moving) and starts collecting all the mist it hits, thereby growing in size. What is the resulting dynamics?

4. You have two identical amounts of water, one is cold (temperature T1) and one is warm (T2). What is the highest possible temperature you can reach by heating the cold water using the heat of the warm water? Only heat exchange is allowed.

5. Take all material constants and represent them as a decimal number in your favourite system of units. What is the probability that the first nonzero digit from the left is '1'?

6. A glass filled to 3/4 with water, on a kitchen scale. You read say 130 gramm. Put a finger into the water (do not touch the bottom). The water is NOT exiting the glass. What will the scale show?

7. A sand clock (hour glass) - all the sand in the bottom part. Measure the weight. Put the clock on a scale with the sand in the top part. The sand will fall down, and for a certain time (say 5 minutes) the scale will be in a stationary state, showing some weight. Is that more, equal, or less than the one from the first measurement?

8. A mathematical dictator imprisones 40 persons. They are told that they will be placed in isolation each, without any means of communication. Random choice will be performed, say regularly once a day, and the chosen person will be brought into a separate 'meeting' room. That room has a switch, which originally is in position off. The visitor can either switch or not. Then after 5 minutes he will be brought back into his cell. And so on forever. Unless any one of the prisoners will declare at some moment, that every prisoner was in the meeting room at least once. If right, they will be all released. If not - heads off. For one night the prisoners can debate about a strategy to get out. Find that strategy, and calculate the average time after which they will be free. Generalize to the case, when originally the switch is an unknown position - either off or on.

9. You are on a cliff, no way but directly down, vertical walls, 100 meters. On the top there is a small tree - for fixing a rope. At 50 meters height above ground, another small tree sticks out of the otherwise extremely smooth walls. You have a rope 76 meter long, just thick enough to carry you. And you have a knife. How can you get down without any artistic nonsense (no swinging, no thinning out of the rope for making it longer ...)

10. You have two inhomogeneous and unequal ropes. But each of them will burn for exactly one hour, if ignited at one end. Use them in order to measure a time unit of 15 minutes.

11. 100 persons are in a queue, ordered by height. The taller ones look in the direction of the smaller ones. each carries a hat - either white or black. No one knows for sure which color he has (say they had to close their eyes when the hats were put on by someone else). The tallest begins. He can say either 'black' or 'white'. Then the next one does similar, etc. At the end we compare - how many spoken words coincided with the colour of the hat of their owner. Question - if the 100 persons were allowed to choose the optimum strategy, what is the maximum number of hits they can reach? Find that strategy. Generalize to N people, M colours (and correspondingly M words which can be used).

12. 100 cups on the table, numbered. We turn all of them (down), then each second (up) again, then each third one, etc, until 'each' hundredth - i.e. only the last one. How many (and which) cups will be NOT standing in their usual operating position?

13. A lion and a man in a large arena. Lion is hungry. Man wants to live. Both are approximated as points. Both can move in arbitrary direction (but bounded by the arena size). Velocities of both can be anything from zero to a maximum velocity - which is the same for both (call it v). Question 1: can the lion find a strategy to run after the man, such that he will get him in a finite time? Question 2: can the man find a strategy such that the lion will never catch him in finite time? Question 3: assume (a) lion is superintelligent (any strategy, whatever complicated, can be used), and (b) lion is performing local optimization, i.e. knows the position and velocity of man, and moves within time dt such that to keep distance to man minimal. Answer both Q1,2 for these cases.

14. Equation a^x = log_a(x), a=1/16 . How many roots?

15. Function f(x) = x^(x^(x^(....infinite number of times .... )))))). For which values of x is the function taking finite (and unique) values?

16. x_{n+1} = x_n^2 + 2. Write as x_n=f(x_0,n) ?

17. Pokerplayer X has five cards. We obtain the information, that one of them IS an ace. We calculate the conditional probability p, that given X has ONE ace, he has at least TWO. Now we assume, that it does not matter which ace he had in the first place. We assume therefore, that we obtain the information, that X HAS ace pique. And now we calculate the conditional probability q, that X has at least one additional ace. Which number is larger - p or q? (modified, from a book by J. E. Littlewood)

18. The light from a light tower reaches a distance R away. The light beam is rotating with constant angular velocity w. A boat is located at a distance R from the tower. It can go with a constant velocity V. I) find the minimal velocity Vm, which allows the boat to approach the light tower without being caught by the light beam. II) Obtain the trajectory of the boat for that case. III) Calculate the time the boat needs to reach the light tower in that case.

19. There are 100 coins on the table - 20 heads, 80 tails. The light is off, you can only feel the coins by touching, but can not distinguish heads from tails. The task is to separate the 100 coins in complete darkness into two groups (that should be fairly easy). But - each of the groups should contain the same amount of heads! You are allowed of course to turn any coin around, but in the darkness you never know, what that means ...

20. There are 12 coins, one is false (either lighter or heavier, we do not know yet). You have a scale. Determine by three weightings, which coin is wrong, and whether it is lighter or heavier.

21. You have a finite squared lattice with period 1 and dimension NxN. You start in the lower left corner, and cut the whole piece with a random pair of scissors, according to the following rules: you cut a distance of one square (length 1) with probability p along the horizontal to the right, or a distance of square root of two along the up-and-right diagonal of a square. And so on. Finally you will cut off a piece of paper. Calculate the average area of the obtained piece.

22. When you accelerate a car, the front goes up, and the back down. When you brake, the opposite happens. Why? Can you construct a car which does everything the other way around?

23. What is the probability to see two walls of the Pentagon when looking at it from outside on the ground?

24. An airplane has 100 seats, and exactly 100 passengers board, one after the other. Each passenger has a correct boarding pass with its seat number printed on it. Among the passengers there is a VERY old lady, who does not care about the boarding pass, and who is not the last passenger. When a regular passenger boards, he chooses his seat. But when the lady boards, she simply chooses randomly and with equal probability an empty seat. The passengers who board after her choose either their seat if it is empty, or otherwise randomly and with equal probability another empty seat. What is the probability that the last passenger will get his seat?

25. KITCHEN PROBLEMS:(a) A cucumber has 99% water. After some time it dries out a bit and has only 98% water. How much total weight did it loose in percentage? (b) Why do we need less water to prepare more eggs in a steam based egg boiler? (c) Why is ketchup not floating out of the bottle when we open it, but does so if we shake the bottle beforehand? (d) Why do professional kitchens use copper based pans? (e) Why are the Knoedel (Kloesse) when put into water first sinking, but later swimming on top? Why are they rotating when the water is hot enough? Why are they rotating even faster if we open the lid? (e) Why is boiling milk 'running away', buit stays in place if stirred? (f) Why is a teapot sending out a final stronger than before steam 'puff' through its nozzle when the heating is suddenly switched off?

26. That That Is Is That That Is Not Is Not Is That It It Is

what is that? put the right ,.!? etc markers

27. 5c = SQRT(25c) = SQRT($/4) = $/2 = 50c ????????????????

28. Put two thick dots on a piece of paper at a distance of 10cm. Close your left eye, always look at the left dot, and get slowly closer to the piece of paper. You will still recognize the right dot (do not look directly at it, still look at the left dot!) At a distance of about 25cm the right dot will disappear. Why?

29. What will you see if the eye balls are strictly frozen?

30. Normally we take a breath every 4-5 seconds. Take 6-8 deep breaths. After that you will realize that for about 30 seconds you do not need to breathe. Why?

31. Is the nerve signal which causes us to breathe induced by varying concentration of oxygen, or CO2?

32. What will happen if we get pure oxygen atmosphere?

33. Why can we save a person with mouth-to-mouth breathing? Isn't our air full of CO2?

34. How old can a humang egg (cell) get?

35. Given is a set of positive numbers a(n) with n=1,2,3,...,100 . The sum over all a(n) gives 271. Which choice of the number set maximises the product over all a(n) ?

36. A village has 7 houses. How to place them on a planar surface such, that for any triplet of them the distance between at least two from this triplet will be exactly 100 meters?

37. P1 tells P2: I know a family with three daughters, the product of their ages gives 36. P2 tells P1: I do not know the ages of the kids. P1 tells P2: actually the sum of their ages equals the number of the hotel room we are in right now! P2 tells P1: I still do not know the individual ages. P1 tells P2: the oldest daugther is blond! P2 tells P1: Oh, then I know all three ages! What are they?

38. A bicyclist is going every day from home to work along some train tracks. After some days he realizes that approaching trains pass by every 20 minutes, and overtaking trains pass by every 30 minutes. How much slower is the biker's speed compared to the train speed?

39. 100 persons on a ring, numbered in order from 1 to 100. Number 1 holds the KICKOUT stick. It proceeds along the ring in the direction of number 2, say clockwise. Once it reaches number 2, number 2 is kicked out by the KICKOUT stick and leaves the ring. Number 1 proceeds further clockwise and hands the KICKOUT stick to the next person (so far number 3). Now number 3 proceeds clockwise, kicks the next one (nr 4) out, and hands the KICKOUT stick to the next one (nr 5 of course). And so on. Who (person number ??) will remain as the last person on the ring?

40. Find the limit of cos(2 e pi N!) for N going to infinity. Here e is the Euler constant, pi is pi, and N! is N factorial.

41. Find the smallest integer p such that the sum of its digits is divisible by 10, and such that the sum of the digits of (p+1) is divisible by 10 s well.

42. Two men are born on the same day and die on the same day. They have the same mother, and the same father. But they are not twins. How comes?

43. Five mafiosi decide to shoot each other. They are positioned in a room, such that the pair distances are all different. Each of them is pointing the loaded gun at another one. Exactly at 12 they all shoot. What is the minimum number of survivors?

44. A couple of former college roommates met up after many years apart and began discussing their children. The first asked the second for his three kids ages, and the second replied: "The product of my children's ages is 36." The first one replies: "I can't tell their ages!" The second one says: "Ok, here is more information - the sum of their ages is the same as our apartment number in college." The first one replies: "I still can't tell their ages apart!" The second one thinks and says: "Oh, I forgot to mention that my oldest child has red hair." The first one thinks and says: "Thanks, I got them!" But how?

45. Clara has two kids - and one of them is a boy. What is the probability that the second child is a boy as well?

Clara has two kids - and one of them is a boy. But now we also learn, that that boy was born on a Tuesday. What is then the probability that Clara has two boys?

Link 1: 100 problems by V.I. Arnold!