Here's a really unique, tough and interesting math problem that I saw earlier on discord. Any sequence of nested (closed) intervals which tend to zero in length must have precisely one number in all of them, and the problem is about two players taking turns choosing the sub-intervals. Will be posting the answer in about a day.
Interesting math problem
Started by Quartemioric [3375600] on in Science.
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Edit: deleted as misread
Edit2: irrational numbers are uncountably infinite, you could always find a closed range of irrational numbers within another interval that the set's cardinality is at least a third. So Bob can always win? Unless Alice is allowed to pick a degenerate interval as A_1, in which case she wins straight away.
"closed range of irrational numbers" the tricky part is that every interval contains infinitely many rationals, they're dense in R - all intervals look the same. You won't exhaust them at any step of the game. I really like the thinking about choosing a degenerate interval, I didn't spot that, but there's a solution if we force the interval choices to have positive length without appealing to uncountability of the irrationals :P
The answer is 42 😏😆
Let me know if I'm missing something here (I'm appealing to the uncountability of the rationals but in a different way) but I've been thinking about this for the past 30 mins or so and want to put some thoughts out. I do think Bob can always win, but I don't have all the details fleshed out.
In step k, Bob is given an interval by Alice and obviously has many different possible choices, but I want to restrict Bob to just consider the two choices where he selects exactly the first third or exactly the last third. We can call the former choice L and the latter choice R and consider the choices that Bob makes through the entire game. Games from Bob's perspective can be represented as the infinite strings of L and R. For example, the game where Bob chooses to alternate between the left and right choice might be represented as the following string:
LRLRLRLRLRLRL...
Suppose we had two strings S1 and S2 that differed at any place along the string. We can assume that at the place of discrepancy, S1 has the L and S2 has the R. If Bob followed S1 for a game, then xi must be in L, and if he followed S2, xi must be in R. Now, L and R are disjoint, and so the final xi will be different.
A diagonalization argument shows that the set of games Bob can play (and therefore the set of xi he can zone in on) is uncountable, and therefore he's got lots more options than just the measly set of rationals.
I have not figured out a precise strategy for Bob yet, but this is my rough thought process for why I think there is one for him.
That's good intuition - there is a winning strategy for Bob, it's to take a rational in the interval *of minimal denominator* and choose a subinterval not containing it on his turn. So eventually all integers are removed from the interval, then all multiples of 1/2, then all multiples of 1/3...
So the number in the final integer won't be rational. If it was, it has some denominator b, which is removed at some step of the game by this strategy.
Oh that is beautiful. I like that a lot.