Gems from the world of probability and randomized algorithms

Card shuffle

Difficulty level: moderate

We all have played cards at least once in our life time. Each game involving playing cardsinvolves shuffling the cards in the beginning. There are many card shuffling strategies. Theaim of shuffling is to reorder the cards so that any ordering due to the previous game getsdestroyed. Ideally, one would like to achieve a random permutation after shuffling cards.Here is a very simple card shuffling strategy. Every time we pick the card from the topand place it at any of the 52 locations selected randomly uniformly. We need to repeat thisstep. After how many steps on average, would you stop and claim that ou have achieved auniformly random permutation ?

If you enjoyed this problem, then you should ponder over the shuffling strategy shown inFigure 1. Can you analyse it as well ?

Figure 1: A popular card shuffling strategy: cut the pack into two equal parts and thenmerge randomly unidormly.

Can analysis of some card shuffling algorithms get someone a prize in mathematics ? Theanswer is yes. Search for Card shuffling algorithms or Persi Diaconis to have the realfun/inspiration.

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