[This problem is worth 10 points.]
A and B play the following game:
Step 1: Two integers k and n are chosen, and revealed to both players.
Step 2: Player A selects two further integers x and N, with 1 <= x <= N. Player A tells N to B, but not x.
Step 3: Player B now asks A questions of the following form: "Does x belong to the set ... of positive integers", for any choice of set.
B may ask as many questions of this form as he wants.
A must give either a "yes" or a "no" answer to each question. The answers, however, do not have to be truthful.
The only constraint on A is that in any sequence of k+1 consecutive answers, at least one must be truthful.
Step 4: After B has asked as many questions as he wishes, he specifies a set X of at most n positive integers. If x is in X, B wins; otherwise, A wins.
Give a non-trivial condition on n, k, and N such that satisfaction of that condition guarantees that B can win the game, and prove that the condition is sufficient.
