For N less than 2^k, binary digit questions don’t necessarily enforce a yes. The x that is excluded may be larger than N. In that case B learns nothing from asking if it is the true x. Thus the “particular element” of Stan Dolan’s argument may not exist.

The argument seems to work for k not less than 2 if N = 2^k, thus sharpening the assertion of the IMO problem to n not less than 2^k – 1 for k not less than 2. For k = 2 it seems to be similar to Jerzy Wojciechowski’s in #10. Combining this with William Meyerson’s #3 we then know that n = 2^k – 1 = 3 is the minimum for k = 2. From other posts we have n = 1 for k = 0 and n = 2 for k = 1. The question of the minimum for arbitrary k appears to be still open.

First I state an equivalent formulation of the problem in terms of existence of certain binary trees. This provides another way to look at the problem and has the advantage that this problem has been studied in relation to unsatisfiable CNF Boolean formulas and the complexity of certain restrictions of SAT, so I can quote results (including one that is partially mine – sorry).

A finite rooted binary tree is called a (k, d)-tree if
(i) every non-leaf vertex has exactly two children,
(ii) each leaf has distance at least k from the root, and
(iii) every node u of the tree has at most d leaves that are both descendants of u and in distance at most k from u.

We call leaves satisfying (iii) with respect to a vertex u “visible from u”. We call them left- or right-visible depending on whether they are descendants of the left or right child of u (the two children of u are distinguished arbitrarily).

CLAIM: The liar’s guessing game for k and n is a win for B if and only if there exists a (k+1,n+1)-tree.

PROOF: Let’s assume first that B wins and fix a winning strategy for N=n+1. We assume he stops asking questions as soon as he can eliminate one of the n+1 possibilities for the unknown x, that is when A’s last k+1 answers are all false for some possible choice for x. Let us build the game tree from B’s strategy: the root corresponds to the initial position before the first question was answered, its two children correspond to the two possible answers, etc. Leaves correspond to the end of the game when the last k+1 answers eliminate a possible value for x. Let us label each leaf with one of the eliminated values.

For the “only if” part we prove that this is a (k+1,n+1)-tree. Parts (i) and (ii) of the definition are satisfied trivially. Part (iii) follows from the observation that all leaves visible from a vertex u of the tree must have different labels. Indeed, otherwise there would be a vertex u’ of the tree with a left-visible and a right-visible leaf having the same label x. But this would mean that both answers to the question B asks at the position u’ rules out x – a contradiction.

For the reverse direction (the “if” part) we use the observation that B wins if and only if he wins with N=n+1. Assume there is a (k+1,n+1)-tree. Label the leaves with values 1<=x<=n+1 such that all leaves visible from the same vertex have distinct labels. This is possible, one can label the leaves in increasing order of their level (distance from the root). Condition (iii) makes sure we have always enough labels to choose from. Now here is a strategy for B to eliminate one of the n+1 possible values for x: Start at the root. At a vertex u ask if there is a leaf labeled x left-visible from u. If the answer is YES move to right child, for NO answer move to the left child. When B arrives to a leaf it can rule out its label and hence wins the game.

QED

And now the references: Let f(k) denote the largest d with no (k,d)-tree (referred to as f_2 in [GST]). Then f(k)=\Theta(2^k/k) by [G] and f(k)=(2/e+O(k^{-1/2}))2^k/k by [GST]. The lower bound can easily be derived from Lovasz's local lemma, the upper bound is based on an elaborate construction. Because of the shift by 1 this means that the smallest n such that B wins the liar's guessing game for k and n is (4/e+O(k^{-1/2}))2^k/k.

[G] H. Gebauer, Disproof of the Neighborhood Conjecture with Implications to SAT, Combinatorica, to appear, Proc. 17th Annual European Symposium on Algorithms (ESA 2009), LNCS 5757, 764–775. http://www.inf.ethz.ch/personal/gebauerh/NeighborhoodConjecture.pdf

[GST] H. Gebauer, T. Szabo, G. Tardos, The local lemma is tight for SAT, Proc. of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011). http://www.renyi.hu/~tardos/SAT.pdf

The idea still uses the fact that within k questions, we can get player 1 to lie about one of the numbers k times (for instance, the strategy of asking about what the i-th bit is does that trick), and then… how to uses subsequent questions to force player 1 into fewer and fewer options for answer choices.

In short, for this strategy, player 2 follows the de Brujin sequence for bitstrings of length k (B(2,k)) to figure out when he should include a particular number i in the question. For the i-th number, start from the i-th element of the de Brujin sequence, if we see a 1, include it in the set for the question. What this does is that for every contiguous block of questions, and no matter what player 1 answers, he will lie about a particular number $k$ times, and this number rotates. In effect, if we examine all possible strategies for player 1 for this set of questions, the player must follow 1 of 2^k possible sequences.

Then, use the last number (2^k+1) as a trap: throw in this number and follow a particular possible answer sequence for k questions. On the k+1-th question, place the last number to oppose that answer sequence, then player 1 is trapped if he followed that answer sequence. Set these traps one bunch at a time, until all of them are gone. It takes a lot longer than the short answer proposed already, but it seems fun. Should have occurred to me to set the trap up at the beginning. Oh well…

So we just need to construct a strategy for A such that eventually (2-epsilon)^k possibilities remain.