I’m a security guy, and I write down my passwords. Bruce Schneier says it’s okay to do so. Just don’t leave it stuck to your monitor. Keep it in your wallet or purse, which you’re already well-trained to keep secure. The greatest danger from someone guessing your password isn’t logging on to your computer while you’re in the shitter, it’s using your account for remote access to the corporate network, from where they leapfrog to other networks or attack further accounts from the inside by exploiting other vulnerabilities like OS rootkits or routers (the massive TJ Maxx hack involved installing a trojan on the router in front of the database and sniffing all the unencrypted traffic that was, in the database, very securely stored).
We’re in an uncomfortable period where the requirements for a reasonably secure password exceed what can conveniently be remembered, but before we move on to devices like swipe cards or one-time password devices (e.g., RSA SecurID token). In a couple decades we’ll likely be using fobs of some sort, if not eye scanners or rectal probes for far greater security.
Logging in is one way to brute force passwords, but there are many scenarios where the password file can be stolen, and your dictionary passwords tested at leisure without a login throttle slowing it down. There’s also rainbow tables, which are pre-computed hashes of passwords up through 7 characters (hashes being how they’re typically stored). At 8 characters, the size of the rainbow table is prohibitively big, which is why the 8 character minimum.
Windows 2000 had a famous vulnerability where it stored passwords greater than 7 characters in length as a concatenation of a hash of seven characters + a hash of the remaining characters. If your password was ‘password’, it would hash ‘passwor’ and concatenate that to a hash of ‘d’. It’s like they were trying to accommodate the limitations of rainbow tables as an attack vector.