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Some funny problems

The red and black cards (from Kosta)

I have a deck of 32 cards, 16 of which are red, the other 16 being black. The game goes as follow: I shuffle the deck, and then reveal the cards one by one. Before each card, you may say "stop - the next card is red". If the next card is indeed red, I give you one CHF, otherwise you give me one CHF. In any case, the game ends. For convenience, we assume that you must say the above words at least once, i.e., at latest before I reveal the last card. Question: Is there a strategy for you where your expected win is positive?

The Chocolate Bar (from Patrice)

You are teaching a math class with a*b students. In the last class before christmas break, you bring with you a rectangular bar of chocolate consisting of a*b squares. You want to break the bar such that you can give one square to each of your students. At any point of time, you can take a contiguous piece of chocolate and break it along the existing horizontal or vertical "valleys" between the chocolate squares. How many times do you have to break until you are left with a*b isolated squares?

Four Coins on the Table (from Kosta)

There are four coins of equal size on an (infinite) table, forming a square of sidelength 10cm. You are allowed to pick a coin A and jump over another coin B - the coin A ends up on the other side of coin B, at same distance. Formally, it ends up on the unique spot such that B is the midpoint of the line segment from A's old position to A's new position.

Jumping with coins over coins. The color red indicates which coin we move, the color blue says over which coin we jump.

As you have seen, you can make several moves of this kind. Question: is it possible to move the coins such that they lie in a square of bigger size? If yes, how? If no, why not?

The Tournament

I got that problem from Dieter Mitsche. Suppose a group of more than two people make a tournament, i.e. each person plays some game against each other person. If every player wins at least one game, then there are three players A, B and C, such that A won agains B, B won against C, and C won against A. Why is that true?

The Birthday Party

On every birthday party, there are at least two people who know exactly the same number of people (not necessarily knowing the same people). Why is that true? Note that we assume that A knows B if B knows A. Note also that this phenomenon is not unique to birthday parties. Weddings are also a good example. Funerals not so much, since if the deceased was famous, you might know him without him knowing you...

Ants On The Stick

There is a stick of length 1 meter, on which several ants are walking. The along the stick, i.e. not around it. An ant walks at 1 centimeter per second. When it reaches the end of the stick, it drops off and will not enter the stick again.[1] When two ants meet, they both turn around by 180 degree and walk in the direction they came from. How long does it take until all ants have dropped off the stick? What is the maximum time it can take?

Infinite Solitaire

You should know the board game solitaire for one player. If not, this little picture should remind you:


In the beginning, in all holes except one there is a stone. A move consists of jumping with some stone A over a neighbor stone B, into the hole behind B. After this move, stone B is taken from the board. So for example, if there is a stone at (i,j) and (i,j+1), but not at (i,j+2), the player can remove the stones at (i,j) and (i,j+1) and insert a stone at (i,j+2). In finite Solitär the goal is to remove all except one stone.
In infinite Solitaire, the board is the infinite integer grid Z x Z , where the lower half grid is occupied by stones, i.e. field (i,j) contains a stone if and only if j ≤ 0. Here is an example of some subsequent moves (the last move is always indicated by the black stone)


In this example we moved a stone to position (0,4). The question now is if there is a strategy to move a stone to arbitrarily high positions, or is there some limit beyond which we cannot move a stone?

The rectangle problem

You have a rectangle of dimensions w*h, that is partitioned into several smaller rectangles. Claim: if every small rectangle has at least one side that has an integer length, then the same must hold for the whole rectangle. I know two proofs of this, one that is cumbersome and ugly (the one I found), and one that is really easy and beautiful (one that I am embarrassed not to have found)

The Elves and the Evil Wizard

An evil wizard has captured 100 elves, and threatens to kill them all. He says "I will give you one chance. Tomorrow morning, I will line you up in front of me house, one behind the other, face to the front. And I will put hats onto your heads, they will either be white or black. Then everyone of you, one at a time, will have to shout the color of his hat. But nothing more, and no hidden signs. Who guesses his color correctly will be free. Who not, will be killed immediately."

Now, the elves discuss what to do. Quite obviously, the elf that will be last in line won't have any hint what the color of his hat will be. So he has to guess. But when the wizard kills an elf, it will make a loud noise, so the other elves will notice that one of them was killed. Now, you have to invent a strategy that will save at least 99 elves.

The Fuses

You are a wizard and you want to make a potion that must be stirred exactly 15 minutes (well, a couple of seconds more or less do not matter). You don't have a clock. You only have two fuses, each of which burns for an hour. They do not burn at a constant speed, i.e. the first inch of a fuse might take 59 minutes to burn, while the rest burns down in a minute. So you simply don't know anything about how they will burn, just that they will burn one hour. How will you cook your potion?

The Conference

I have this problem from my friend Stefan. There is an international conference, and every country sends two delegates. For dinner, there is a large round table, with a seat for every delegate and a name card on every plate. Now, the delegates talk and discuss and sit down in a random manner. The claim is that you always can rotate the table such that at least two delegates sit at their correct place, i.e. in front of their name cards.

Remarks

[1] It should be noted that an ant does not suffer any damage when dropping off the stick. There is no cruelty against animals on this webpage.