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\begin{document}


\begin{center}
\bigskip {\Large Puzzle 3}

\bigskip 
\end{center}

A king has his birthday. So he decides to let go some of his prisoners. He
actually has 100 prisoners at the moment. They are each in a separate cell,
numbered from 1 to 100. Well, he is a high tech king. He can close or open
any prison door by a single click on the cell's number on his royal laptop.
When he clicks at a locked door, it opens. When he clicks at an open door,
it locks. \ At the beginning, every door is locked. 

\bigskip 

First the king clicks on every number from 1 to 100 (therefore opening every
door). 

\bigskip 

Then he clicks on every second number from 1 to 100, (i.e.2, 4, 6, 8, 10, .
. . ). \ 

\bigskip 

Then he clicks on every third number.(i.e. 3, 6, 9, 12, . . .  ) \ Now he is
opening some doors, locking others.

\bigskip 

Then he clicks on every fourth number. (i.e. 4, 8, 12, 16, \ . . . .)

\bigskip 

Then on every fifth....

\bigskip 

And so on, every sixth, every seventh, etc. 

\bigskip 

Until every 100th; finally, he only clicks on the number 100. Then he orders
that the prisoners that find their door open may go free.

\bigskip 

Who gets to go and who has to stay?

\end{document}