180. Twenty Questions. I recalled an old game that I often played in my youth. Someone thinks of something specific, for example, Big Ben, a knocker on a front door, the chime of a clock in the next room, the top button on a friend's jacket, or Mr. Baldwin's pipe. You have to determine what was thought of by asking no more than 20 questions, to each of which the answer can only be "yes" or "no."

Questions should be asked carefully, as, for example, by asking: "Is it an animal, plant, or mineral?", you might get an unsatisfactory answer of "yes" and thus waste one question. An experienced player of "Twenty Questions" rarely makes mistakes; I know of extremely difficult cases where a solution was still found under such conditions.

Recently, I was offered a new version of this game, which requires some ingenuity, and different people may approach the solution differently. The game is as follows. I think of a six-digit number. Can you guess it by asking only 20 questions, to which I will answer only "yes" or "no"? After the twentieth question, you must name the number.
180. There are different ways to solve this puzzle, but the simplest one, I believe, is as follows. Suppose the six-digit number is 843712.

1) Is it divisible by 2 without a remainder? Yes.

2) Is the quotient divisible by 2 without a remainder? Yes.

3) Is the new quotient divisible by 2 without a remainder? Yes.

Your twenty questions should all be the same. Write down 0 for each "yes" and 1 for each "no" from right to left. After asking the 20th question, you will get 11001101111111000000. This is nothing but our number 843712 written in binary. Since there are 6 zeros on the right, the first 1 from the right represents \(2^{6}\), the next one \(2^{7}\), and so on. By adding all the powers of two from \(2^{6}\) to \(2^{15}\) and adding \(2^{18}\) and \(2^{19}\), you will get the number 843712 in decimal notation.

If the number is not too large, for example, 100000, then 17 questions would be sufficient if you knew that the quotient is 0. The last three questions would add three extra zeros in the higher places of your binary number. To avoid confusion, it is better to assume from the beginning that 0 is divisible by 2 without a remainder, and the quotient is 0.