Andreas Lundblad
Dec 11, 2016

Password Constraint Puzzler

I was creating a Java snippet for password generation for programming.guide and stumbled across a fun puzzler. Suppose you have the following scenario:

You’re a user trying to register a new account. The site says that the password must be at least 8 characters, so you enter "qurcoehx". The site responds with

   Password must contain at least 1 digit

You’re lazy and don’t want to type more characters than necessary, so you stick to 8 characters and replace the c with a 7: "qur7oehx". The site accepts the password.

The puzzler: Is the second version “safer” than the first version? Or, put differently: Are there more 8 character passwords with exactly 1 digit, than there are 8 character passwords with only lower case characters?

On the one hand one might think “a mix of characters and digits must be better than just characters”, but on the other hand we have “being forced to ‘downgrade’ a character (26 options) to a digit (10 options) must surely be bad”.

So which one is it?

The solution becomes apparent if we consider an 8 character password with 1 digit as

  • a 7 character password (267 choices),
  • 1 digit (10 choices), and
  • a digit position (8 choices)

We see immediately that 267×10×8 > 268. What saves us here, intuitively, is the fact that the digit has many possible positions.

Conclusion: It is indeed “safer” to replace a character with a randomly selected digit.