239. Twenty Questions. I want to offer you a variation of the well-known game of "twenty questions," which works as follows. I think of a natural number, and you have to guess it by asking no more than twenty questions, to each of which the answer can only be "yes" or "no." What is the largest number I can think of so that you can still give me the correct answer after twenty questions?
239. If one object is chosen from a finite set of objects, the best way for the questioner to determine it is as follows. At each step, it is necessary to find out whether the given object has a property that is possessed by exactly half of the "suspects." In this case, the number of "suspects" is reduced by exactly half with any answer. By applying such a method, the questioner can, by asking 20 questions, determine any natural number not exceeding \(2^{20}\).

For example, on the \(i\)-th step ( \(i=1,2, \ldots, 20\) ) one can ask the question: "If the number is written in binary, will its \(i\)-th digit be equal to 1?" Note that if we receive the answer "no" to all questions, then the sought number is 21-digit (in binary) and consists of 1 followed by 20 zeros, that is, it equals \(2^{20}\). If at least one of the questions is answered with "yes," then the guessed number is at most 20-digit and is fully determined after 20 answers.

[X. Gehman, A. M. M., 58, 40 (January 1951).]