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This is a collection of some nice puzzles that I have come across.
There are ten wires that run underground across a river. Thus on each
of the banks, there are ten ends of the wires. Unfortunately, given an end
of a wire on this bank, one doesn't know which end on the other bank it
corresponds to. Suppose that you an ohmmeter and a bunch of wires. (You can
then, for example, connect two ends on side A of the river, and go to the
side B and measure the resistance between every pair of ends to figure
out which pair on side B corresponds to the pair that is connected on side A).
What is the minimum number of crossings you need to make in order to figure
out the matching ends of all the ten wires?
From a random distribution on the plane a point (x,y) is chosen. It is know that x and y are not equal almost always (in other words, probability that x=y is zero). You are then given one of these two numbers randomly. You have to guess whether the number given to you is the smaller or the bigger of the two. Come up with a strategy which gives the right answer with probability (strictly) more than half. Your strategy should work irrespective of the distribution.
Note that if for example you toss a fair coin and say whether the given
number is bigger or smaller than the other, then the probability that your
answer is correct is exactly half.
A professor decides the following grading scheme in his class. After
the final exam is graded, he keeps all the papers upside down on his table
in a random order so that no student can recognize his own paper. Each
student during his turn can overturn at most n/2 of these papers
(where n is the total number of students in the class) and guess whether
he received an A or a B on the final (there are only two grades given).
Obviously the student doesn't know which paper is his, so it is not
gauranteed that he will find his own score by looking at n/2 scores.
The papers are then turned back and kept in the original order.
The students cannot pass any information to others. All the students
pass the course if *everyone* guesses their grade correctly, and they fail
otherwise. Come up with a strategy that the students can decide on
beforehand, so that the probability that they all will pass is more than
a positive constant independent of n.
In each of the three following figures, there is a smooth curve C and two directed straight lines A and B. In each of these three cases, is it possible to smoothly translate the line A to B (with directions appropriate) so that the line is never tangent to the given curve C?
In a town, every club has an odd number of membership and every pair
of clubs has an even number of common membership. Show that the number
of clubs is less than the number of people.
Consider a polytope in R^n whose vertices lie on the unit sphere. Show
that the spheres whose diameters are the line segments connecting the
vertices and the origin, cover the entire polytope. For example, in
dimension 2 and in case of a triangle, this reduces to saying that if O is
the circumcenter of a triangle ABC, then the circles with diameters OA,
OB and OC cover the entire triangle.
The space R^n (n copies of the reals) has 2^n octants. If we are given
2^n vectors belonging to the interior of distinct octants, show that they
will span the whole of R^n.
Three people A,B and C are locked up in three different cells in the
hell. God gives them an opportunity to get out of hell. He calls these
three one by one in a random order and shows them a switch. This switch is
off to start with (though this information is immaterial). They are allowed
to operate the switch from off to on or from on to off, or do nothing. If
they can say correctly that all the three people have seen the switch at
least once then they are free to go. Given that God calls each of them
infinitely many times, find a strategy for all the three of them to get out.
As an example, God can call the people in the following order:
ABCABCABCABC... Obviously, your strategy should work no matter what the
order of calling is.
An n by n square grid is filled with real numbers such that the number
on each square is the average of the numbers in the neighbouring squares
(two squares are neighbours if they share at least one common vertex). Show
that all the numbers are the same.
There are n doctors and m patients and each doctor needs to operate each
patient. And to operate a patient the doctor has to wear gloves. What is the
minimum number of gloves required so that no two people touch the same side
of the gloves? You can start with the simplest case where n=m=2 and the
answer in this case is "two".
There are hundred people standing in a line, each wearing a blue or
a red cap. Each of them can see the colour of the hat that the people in
front of him/her are wearing. They are supposed to guess the colour of
their hat and if they guess it wrong then they are dead! Obviously, they
cannot pass any information among themselves, but they can always have
a strategy decided before hand. The question is - how many people can
Two cities A and B are 100 miles apart. A person has to travel from
one city to the other in his old old car, which can travel only 50 miles
with its tank full. Unfortunately, there are no gas stations between the
cities. (Of course it is dumb, but still) suppose that the person cannot
carry any extra gas in his car. What is the minimum number of refuellings
at A does he need to do in order to reach B? (The whole point is that he
can store fuel in between A and B and the problem is to figure out a
clever way of doing it).
There was this crazy lottery in which it is ensured that at least
90% of the participants win a prize. The way this is played is as
k participants enter a room wherein the conductor of the lottery throws die. If the outcome is 6 and 6 then all these participants in the room win a prize and the lottery ends, and otherwise next set of participants are allowed to come into the room after vacating the room. Note that the number of participants inside the room, that is k, is a variable.
The way they ensure that at least 90% of the participants win a prize is setting k = 1,9,90,900,... (Convince yourself that this works).
One day, you see in the newspaper a list of names, with "90% of these
people won a prize in a lottery" written underneath. Your friend's
name is in that list. Assuming that you don't know anything about the
lottery, what is the probability that your friend won a prize? What
if you already know that this lottery is the one described above?
Two persons A and B are given the following task:
From a pack of (52) playing cards A should take 5 cards randomly (without replacement). He then should pass four cards to B, one by one. B should then be able to tell what the fifth card is.
What protocol can A and B use? Can you generalize this?
You have a glass cube with a hole at the center of a face. How can you
fill the cube with water occupying one third of its volume? (there is
no scale provided).
Once upon a time, there lived a king who had two sons. These two sons were not only twins by birth, but also equal in all aspects. When it was time to decide the successor, the king called his two sons and assigned them a task. The task was to take their horses to a place nearby and bring it back. The person whose horse comes the last will become the successor.
With this task in hand, the two sons set out, obviously, as slow as possible. Days pass by, and the sons don't move more than couple of meters per day! None of the two wants to loose, but certainly they want to be done with this contest. On one fine day, an old man meets these two, on his way to the capital. Hearing about the "race", he gives them an advice. At once, the two sons run to the horses and start riding them as fast as possible. The result of the race and the remaining story is irrelevant.
The obvious question : What was the suggestion by the old man?
As usual, a dead man has the task of finding his way to heaven. There are
three doors A,B and C in front of him, two of which lead to hell and the
other to the heaven. God asks him to choose a door to enter and he chooses
door A. Then the God, who always speaks the truth, tells the dead man that
the door C leads to hell. He is then asked if he wants to change his mind.
What should he do?
There are five looters who loot 100 gold coins. After looting they decide to split up and they get into a controversy regarding the share of the loot each person gets. They decide to do the following:
A lottery will be held to number each of them from one to five. Then the first person will come up with a sharing formula and there will be a voting on it. If there are at least half the people (three in this case) who agree with the sharing system, then they share according to it. Otherwise, they kill the first person and the second person will come up with another system of sharing the loot among 4 people. This continues until they share the loot.
What should the first person's sharing system be?
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