What are the ten questions to determine a number between 1 and 1000?

Answer 1

You divide the range, 1-1000, into two smaller ranges, of exactly the same size, or a size difference of only one. At first, one range is 1-500, the other 501-1000. You ask, for example, "Is your number more than 500?" The answer places the number you must guess in one of the ranges. Let's assume the other person replied "yes", you once again divide the remaining range into two smaller ranges - 501-750, and 750-1000. So the next question is "Is your number more than 750?" Depending on the answer, you again divide the remaining range into two smaller ranges. Since 210 = 1024, you can guess any integer between 1 and 1024 with only 10 guesses; any number up to about a million with 20 guesses, or any number up to about a billion in 30 guesses, by using this method.