Möbius strip

This is interesting puzzle that gives unexpected results for people unfamiliar with properties of Mobius strip. A model can be easily made from a paper strip,  half-twist one end and then joining the ends of the strip together with scotch tape to form a single strip. 

It has few interesting properties, one of which is that mobius strip has only one surface. For example, if you start drawing line from the seam in the middle of strip, it will meet the starting point and will be double the length of the original strip of paper.

Now, here is questions:

What will happen if you take scissors and cut mobius strip along the middle (or follow the line drawn as explained in the example above)? What will happen if you do it again ?

It is easy to find answers to those questions by trying out, but try to solve in your head first and then check your solution with experiment. 

Personal note, my son was amazed with results and which prompted this particular puzzle.

Pants or no pants

On last episode of MathFactor podcast there was an interview with Prof Ed Burger of Williams College where he discussed interesting puzzle:

Imagine that you have a rope, one end of which is tightly tied to your left ankle and another to your right ankle. So rope and your legs form a loop. Is that possible to remove your pants (or shorts) then turn them inside out and put them back on (in the "inside out form") without breaking the loop ?
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Note: Comments may have spoilers! Come up with solution on your own before reading them.

Light bulb and 3 switched

This one is old, I do not recall where I picked it up from.

There are 3 switches at one end of the long corridor and small room behind closed at another end. Room has a conventional light bulb controlled by one of the switches. You start with all switches been off. How many trips to that closed room is needed to find which switch is actually controls bulb?

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Note: Comments may have spoilers! Come up with solution on your own before reading them.

Number sequences

Here are couple of interesting number sequence puzzles I came across recently.

What number is next in the series?
1 11 21 1211 111221 …

What number is missing ?
823543 46656 3125 256 … 4 1

I think second one is much simpler for people with math aligned skills or engineers while first one is more think out of the box type of problem.
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Note: Comments may have spoilers! Come up with solution on your own before reading them.