[BOTW]

If you were to provide the classes of items, I'm sure you could set up a good guessing strategy.

It looks like her strategy is better than yours, but it depends on how things are divided up and the sizes of the sets. If she goes through and guesses the smallest set, each of the 24 names, she gets at max 6 guesses before your win, so she only wins 25% of the time.

However, assume you divide in half and she can divide in thirds (until there are only a few items left or she sees she needs a new strategy because you are about to win).

If you haven't already won, she wins (or ties) 1/3 of the time at every step along the way; 2/3 of the time she moves on to the bigger remaining set.

That is, in the first step, she has either 8 items to choose from (for a win/tie) or is left with 16 (you have 12).

In the second step from 16 items, 1/3 of the time she has 6 items (tied with you), or falls behind with 10. (So, 1/3 + 2/9 = 5/9 times she is winning or tied with you now.)

So, how often does she get behind in the first two steps? 2/3 * 2/3 = 4/9. It appears that she gets behind (2/3)^n times in n steps, or the more steps the binary search takes, the more likely she is tied or winning.



Looking at the guessing numbers problem (and rounding), binary guessing of 50, then 25 or 75 etc. looks like it will take a max of seven steps.

She guesses 66 (guess one). 1/3 of the time she has 33 numbers left, 2/3 66 left.

If it is 33, then she at least ties, (binary is at 50).

If it is 66, then it will be 22 (at least ties) or 44 (guess two) (binary is at 25).

If it is 44, then it will be 14 (at least ties) or 30 (guess three) (binary is at 13).

If it is 30, then it will be 10 or 20 (guess four) (binary is at 7).

If it is 20, then it will be 7 or 14 (guess five) (binary is at 4).

If is is 14, then it will be 4 or 10 (guess six) (binary is at 2).

If it is 10, then it will be 3 or 7 (guess seven) (binary is at 1 and done).

So, she falls behind at guess four in the worst case. How often does she fall behind binary guessing? 2/3 * 2/3 * 2/3 * 2/3 = 16/81 ==> guessing in thirds is tied or ahead the majority of the time. (Or I'm too tired to think straight, someone correct me.)