Need to gain a trit (ternary information) with each question, so we must use possible silence in return to the questions. Generate random information somehow, and keep it hidden; e.g. flip a coin secretly.

First question is "Is the number in {1, 2, 3}, or is it in {4, 5, 6} and did I flip heads?", with "and" binding stronger than "or". Use a similar second question to weed out the number from the appropriate group of three.

Consider the space of functions N -> {red, blue}, called F. The contestants define an equivalence relation ~ on F as follows. Two hat distributions f, g: N -> {red, blue} are equivalent iff there exists a natural number n with f(k) = g(k) for all n >= k. (I.e. the hat distributions differ only on finitely many elements smaller than some n.)

Take the quotient set F/~, i.e. the set of all equivalence classes. By axiom of choice, the contestants agree on a choice function c: F/~ -> F, such that for any class B of equivalent hat distributions, c(B) is a certain distribution in B.

During the actual contest, each contestant determines the equivalence class B of the hat distribution at hand. Ignoring/changing a finite number of hats cannot change the equivalence class of any hat distribution, so each contestant, no matter how far up in the sequence, is able to determine B. Each contestant n now guesses c(B)(n). By definition of ~, only finitely many candidates get it wrong. (The probability of each candidate being right is still 1/2, so you can't turn this into a measure space.)