Why is 6 "ugly"? Each number k for which the player playing at that point will win leads to 2k being a loss, meaning that 2k + 1 is a win. Whereas if k is a loss, then both 2k and 2k + 1 are wins. So part a) is really easy, because we can just repeatedly halve the numbers to figure out what the values should be, and part b) is also easy, because our proportion of wins to losses in the limit is described by a simple markov chain.
[edit: another minute's thought... actually, the obvious implication of the above is that every odd number is a win, and therefore losses are only those numbers with odd numbers of 2s in their prime factorizations]
In any case, I don't see why these questions belong on Hacker News. They're the sort of fun puzzles that make for a good high school math camp quiz (a clever high school student should figure most of them out in an afternoon), but none of them contains any deeper insight or relevance, as far as I can tell.
My friend's son is an aspiring MathCamper so I thought I'd post the type of questions they use to winnow the applicants, in case anyone was curious and/or wanted to exercise their proof muscles.
[edit: another minute's thought... actually, the obvious implication of the above is that every odd number is a win, and therefore losses are only those numbers with odd numbers of 2s in their prime factorizations]
In any case, I don't see why these questions belong on Hacker News. They're the sort of fun puzzles that make for a good high school math camp quiz (a clever high school student should figure most of them out in an afternoon), but none of them contains any deeper insight or relevance, as far as I can tell.