How many "Yes or No" questions are needed to guess any 7 digit phone number?


 

Solution to Problem:

The answer is 24 questions.

Use what is known as a "binary search" to ask the questions.
Your first question would be:
Is the number greater than 5000000?
This would divide the 10 million possible phone numbers (from 000-0000 to 999-9999) in half with just one question.
Each succeeding question would divide the remaining numbers in half.
So we are looking for the solution to the equation 2x = 10000000

You can solve this with logarithms:
x ln 2 = ln 10000000
so x = 23.25
therefore, it would take 24 questions to be absolutely sure that you would be able to guess any 7-digit phone number.

OR, of course, you could just examine
224 = 16,777,216 and 223 = 8,388,608 to see that 24 questions are needed.




Correctly solved by:

1. Tom Kelley Winchester, VA
2. Joe Vance Winchester, VA