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Puzzles

On this page I post some nice problems, which do not require any deep mathematics, but which I liked a lot. Some of them have beautiful solutions, some have surprizing statements. None of them is my own problem, and I mention the source where it is possible.

This section is supposed to bring some fun, to let somebody feel the inspiring beauty of mathematical idea. If you solve the problem, you know what I mean.

I do not post solutions here. If you want to know one try to ask me. If I have one I will definitely share it.

Also, if you enjoyed some other problems,
which are easy to formulate and not very easy to solve, I will be glad to try those.
You are more than welcome to share them here:

So, here are the problems:

**1.** *Does there exist a closed chain, consisting of 6 segments, which passes
through all vertices of the cube?*

This problem was given on the Soros olympiad in Ukraine ~1996

**2.** *Given a point A in the space can one place several opaque right-angled
parallelepipeds in the space without intersections in such a way, that you can not see any
corner of any parallelepiped from point A?*

This problem was given on the Tournament of the Cities ~1996. Author: Alex Belov

**3.** *Rockclimber stands on the top of the cliff of height 100m. He has the rope
of length 75m and a knife. There are only 2 places on the cliff where he can attach the rope: the very
top point, and on 50m height. Can he survive and get down?*

This one is fairly easy problem

**4.** *There are 15 stones in few piles. Take one stone from each pile and form a
new pile from these stones. Repeat, repeat, repeat,... Prove, that at the end you will have 5 piles
containing 1,2,3,4 and 5 stones correspondingly.*

**5.** *There are 100 prisoners in the prison, which are located in separate rooms
and do not see each other. The government gives them the chance to obtain the freedom. They tell, that
every day, starting from January 1, one prisoner will be chosen at random and put into a special room with a lightbulb and a switch,
where he can not do any marks, but he can either change the position of the switch or leave it as it was.
After he spend this day in this room they move him back to his own room and next day choose one more
prisoner randomly (the same person can be chosen twice or more), and so on. At any moment of time any of
the prisoners can claim: "Everybody visited this special room!". If he is right they are all free, if not,
everybody will be executed. Can they come up with the strategy that will guarantee life for them and freedom with probability 1
after everybody was there?
*

**5.** *(!!! Very different from the previous one !!!) There are 200 prisoners in the prison,
which are located in separate rooms
and do not see each other. The government gives them the chance to obtain the freedom.
To each prisoner they assign a unique number from 1 to 200. Then they say to everybody
They tell, that
every day, starting from January 1, one prisoner will be chosen at random and put into a special room with a lightbulb and a switch,
where he can not do any marks, but he can either change the position of the switch or leave it as it was.
After he spend this day in this room they move him back to his own room and next day choose one more
prisoner randomly (the same person can be chosen twice or more), and so on. At any moment of time any of
the prisoners can claim: "Everybody visited this special room!". If he is right they are all free, if not,
everybody will be executed. Can they come up with the strategy that will guarantee life for them and freedom with probability 1
after everybody was there?
*

In case you want to solve more puzzles here are few links to other pages

- Easy with answers:

http://www.frontiernet.net/~fys/puzzles.htm

Paradoxes and "Paradoxes"

On this page I post some nice problems, which do not require any deep mathematics, but which I liked a lot. Some of them have beautiful solutions, some have surprizing statements. None of them is my own problem, and I mention the source where it is possible.

Updated:
August 21st, 2012