Puzzle in Martin Gardner book What is the official name of this problem? Martin Gardner gives introduction in his book "Math circus". The problem belongs to 1D random walk. What can be read to gain deep insight into this problem? Or other useful resources. 

We can complicate matters by allowing transition probabilities
  to vary from 1/2 and by allowing steps longer than one
  unit. Consider the following curious paradox first called to my 
  attention (in betting terms) by Enn Norak, a Canadian mathematician.
  A walker starts 100 steps to the right of 0 on a line
  that has no barriers. Instead of a coin a packet
  of 10 playing cards-five red and five black-is used as a randomizer.
  The cards are shuffled and spread face down and any
  card is selected. After its color is noted it is discarded. If it is red, the walker steps to the right: if black, he steps to the left.
  This continues until all 10 cards have been taken. (The transition
  probability varies with each step. It is 1/2 only when there
  is an equal mixture of red and black cards before the draw.)
  The walk differs also from walks discussed above in that before
  each card is noted the walker chooses the length (which need
  not be integral) of his next step.
  Assume that the walker adopts the following halving strategy
  in choosing step lengths. After each card is noted he takes a step
  (left or right) equal to exactly half of his distance from 0. His
  first step is 100/2 = 50 units. If the card is red, he goes to the
  150 mark. His next step will then be 150/2 = 75. If the first
  card drawn is black, he goes left to the 50 mark, and so his next
  step will be 50/2 = 25. He continues in this manner until the
  tenth card is noted. Will he then be to the right or to the left of
  the 100 mark where he began the walk?
  The answer is that he is sure to be to the left. This may not
  be very surprising, but it is surely astonishing that, regardless 
  of the order in which the cards are drawn, he will end the walk at 
  exactly the same spot.

 A: Here's another expression of the same trivial puzzle. I buy some shares for $1000.  In five of the next ten weeks (I'm not saying which five), their price rises by 50%; in the other five, it falls by 50%. What are my shares now worth?
A: Sure, multiplication is commutative, but there is more to it than that. While being reasonably easy, this puzzle suggests variations in ways that the equation $x\cdot y = y\cdot x$ doesn't.  
In his wonderful paper Games People Don't Play, http://www.teorver.ru/newkatalog/1193689162.pdf, Peter Winkler describes essentially the same game as ``Next card color betting'' (a bit of googling also turns up http://www.maxdama.com/?p=137 and http://www.dartblog.com/data/2008/08/007950.php). But there the player, Victor, wants to end up as far to the right as possible (increasing his bankroll). It turns out that there is a strategy that guarantees him to end up with $2^{10}/\binom{10}{5} = 256/63$ times his initial bankroll, or about 406 steps to the right of the origin.
This is a game that children can understand, but if we pursue the analysis, it doesn't stop until we have developed, besides insights into hedging strategies, a good deal of nontrivial mathematics including information theory (the amount of information Victor has about the red-black sequence dictates exactly how much money he will ideally make by betting), the Wallis product formula (showing that his final bankroll is asymptotically $\sqrt{\pi n}$ for a deck of $n$ red and $n$ black cards), and even the central limit theorem.   
