Math CentralQuandaries & Queries


Question from Tim, a student:

pretend I have a password that is - 9k7d4112p5o

assuming that you continuously guessed wrong until that exact combination was the only possible answer left; how many times would you have to guess?

capital letters are not relevant and you do not know how many digits there are

Hi Tim. I have to make some assumptions to give you a numerical answer.

First assumption: password characters can be any English alphabetic character or any numeric character (no symbols, spaces, etc). So that means there are 26+10 choices for each character.

Second assumption: Your password is as long as it is allowed to be. That means that since 9k7d4112p5o is 11 characters long, then passwords must be at least 1 character and at most 11 characters long.

There are 36 possible one-character passwords.
There are 362 possible two-character passwords.
There are 363 possible three-character passwords.
and so on.

So the next question is this:
What is 36 + 362 + 363 + ... + 3611 ?

I won't show you the derivation (you could just use your calculator too), but the sum like this can be simplified to:


So the number of guesses is 36(3611 - 1)/35 which my calculator says is 135 382 323 952 046 196 or about 135 quadrillion. That would take some time.

Steve La Rocque

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