#DIGITAL COIN FLIP CODE#

Ī few years ago, I tried to present this problem quickly, and ended up getting the math wrong. The fraction 22/7 is a fairly good approximation for, so the odds of a win are about 1/. One 4-flip game with strange odds is HHHH vs. Every game tends to beat the game in front of it.

Notice the clockwise ring of arrows on the outside. HHHH and TTTT are losers, from an odds standpoint. Repeating all of this for the 4-flip games, here are the 14 games with non-obvious strategies. With results from games of 200 flips or less, the odds can be determined very precisely. From that, many more terms were calculated.

#DIGITAL COIN FLIP CODE#

The sequence from the brute-force code leads to a generating function, which can be used to find many more terms in the sequence.Īgain, the start of a sequence was found by brute-force methods, and that sequence led to an elegant function. With the sequence, FindSequenceFunction or FindGeneratingFunction can be used to get the underlying function. A head and a tail are added to all games without a winner, and then the process is iterated. The set of flips that could have a winner is calculated, and then all the winners are removed. The sequences of a game can be calculated with the piece of code in the downloadable Computable Document Format (CDF) file. The corresponding number of wins is the beginning of the Fibonacci sequence. Here are the ways HHH can win after 3, 4, 5, and 6 flips. For any game your opponent picks, you can pick a game that beats it, making this a nontransitive game. Calculating these odds can be both tedious and mathematically demanding-a natural job for Mathematica or Mathematica Home Edition.Īs a picture, here are the 6 games that are not obvious losers. Here is a table of odds and facts for various 3-flip games. This is the strange world of Penney’s game. Which one? The odds of the event occurring are given at the end.

Phrased a different way: Suppose I offer a bet on a series of coin flips. TH, HH will win if the first two flips are HH and will lose if any of those flips are tails. Beforehand, the players each pick a sequence of flips.

Digital coin flip