Generalized Superposition of Coins Problem

[ニア愛 !pQsULI4sXc No.12801767]

Given two natural numbers n >= 4, and k < n, consider the following two player game G(n,k):

Alice and Bob stand around a spinable table. Spaced evenly near the edges are n indistinguishable small boxes with a coin inside each box. The coin has face-up value H or T. The boxes are initially all closed meaning that the values of the coins are not visible.

On Alice's turn, she:
- opens any k boxes of her choosing
- sees the value of each coin
- sets them in any way she'd like (e.g. if k=2, she could set them as HH, TT, HT, or TH)
- closes those k boxes

On Bob's turn, he:
- has Alice leave the room so that she may not see what happens
- opens all of the boxes inspecting the coins, but doesn't touch anything.
- closes all the boxes
- spins the table to any new orientation of his choosing
- has Alice re-enter the room

They take turns in an alternating fashion, with Alice to start. The game ends and Alice wins if all coins are H or all coins are T at the start of Alice's turn. As such the game may never terminate.

Let F(n) be the smallest k such that Alice has a strategy to always win the game G(n,k). Find F(n) and prove that your solution is correct.

[ニア愛 !pQsULI4sXc No.12801786]

Some warm ups and pointers:
It's pretty easy to see that G(n,1) never has a winning strategy for Alice. (remember n >= 4)

You can then show that F(4) = 2 by finding the winning strategy of G(4,2).

I'd also recommend figuring out F(5) and F(6)