how much money you need to have to guarantee a win
is bad because you don't describe the algo which will be used.
You see, there can be a smart one, and the worse one.
Smart one can save up you money, and yeah, it makes sense to use it, but you don't specify which one is gonna be used !!
Because, even in case of using the worse algo there also can be a challenge to programmer to find out the amount of money.
to give you an idea what do I mean with algos, here are the examples:
Numbers are [1, 5], and the pick is 4. Worse algo would guess 3 and then 5 (spent money is 8).
Numbers are [1, 5], and the pick is 5. Smart algo would guess 2 and then 4 (spent money is 6).
Please, update the description.