Some Puzzles

You, my friend, are about to witness the best card trick there is.
Here, take this ordinary deck of cards and draw a hand of five cards from it.
Choose them deliberately or randomly, whichever you prefer  but do not show
them to me! Show them instead to my assistant, who will now give me four
of them: the 7 spade, then the queen heart, the 8 club, the 3 diamond.
There is one card left in your hand, known only to you and my assistant.
And the hidden card, my friend, is the king of spade.
Incredible! How is this possible?
(for a reference see: M. Kleber, The Best Card Trick,
The Mathematical Intelligencer 24 (2002) 911)

Suppose you've got a nice Kandinsky painting with a string attached
from the top left hand corner to the top right hand corner.
In addition you have two nails. How can you hang up the painting
with the two nails such that it will fall down if either of the two
nails is removed?

Suppose you are given 12 balls and a scale which can only measure
relative weights. One of the balls has a different weight from the
others (but it is not known whether that ball is lighter or heavier).
How can you determine which of the balls has the different weight
in just three measurements?
Can you also do it with 13 balls?

What is the missing number?
10,11,12,13,14,15,16,17,20,22,24,31,100,?,10000,1111111111111111
 Suppose you have two sticks and some matches. You only know that
each stick burns in one hour. How can you measure 3/4 of an hour?
 At a high school, there are 1000 lockers and 1000 students. At the
beginning of the day all lockers are shut. Student 1 comes in and opens
all the lockers that are divisible by 1. Then student 2 comes in and
changes the state of all lockers divisible by 2. Student 3 changes the
state of all locker doors divisible by 3 etc.. How many doors are open
by the time student 1000 is done?
 Or you might try the The Hardest Logic Puzzle Ever
I hope you are not as desperate as this person!
If you know some other puzzles or have questions feel free to email me at
anne@math.ucdavis.edu.
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