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Hint to puzzle 146: Odds and evens

  1. How many different possible games are there?

    Let g(n) denote the number of different possible games with upper bound n.  Express g(n) in terms of g(n − a) and g(n − b), where a and b are constants to be determined.

  2. How many different possible games of length k are there?

    Let {1 less than or equal to a1 < a2 < ... < ak = n} denote a game of length k.  Consider the sequence defined by bi = ai − i + 1.