The coin-flip sequence game you can always win second

Question

We flip a fair coin over and over, recording heads (H) and tails (T). First you announce a length-3 pattern (say HHH). Then I announce a different length-3 pattern. We keep flipping until one of our patterns appears as three consecutive flips; whoever's pattern shows up first wins.

Going second sounds like a disadvantage. Show that I, moving second, can always pick a pattern that beats yours.

Accepted Answer

Going second is a huge advantage here because "which pattern appears first" is a non-transitive relationship, there is no single best pattern, so whatever you commit to, I can name something that beats it. The winning recipe (for the second player). Take your three symbols. My pattern is: the opposite of your middle symbol, followed by your first two symbols. If you say H H H, your middle symbol is H, so I lead with T, then copy your first two (H H): my pattern is T H H. Why T H H beats H H H. Your pattern HHH only ever occurs if the run starts cleanly with heads. But consider the first time …

The coin-flip sequence game you can always win second

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