One Hundred Locks
Somewhere, deep beneath the pentagon, there is a hall of 100 locks which control a secret undergound computer. There are also 100 keys, held by 100 officers (talk about pork barrel!) The keys (and for convenience the officers) are numbered 1-100. The first key, held by the first officer, works in every lock. The second key, held by the second officer, works in every other lock. The third key, held by the third officer, works in every third lock. And so on... The 100th key, held by the Vice President, works in only the 100th lock.
In the event of a national emergency (this computer is so secret, no one even knows what kind of emergency it could be used for) all 100 key officers line up, and, one by one, walk down the hall and try their key on the locks. Of course, if two people turn their key in the same lock, it will end up in the position it started in- one will unlock it, the other will lock it again.
The lock system is setup so that if the entire procedure goes correctly, the locks will end up in the correct positions to activate the computer (you see, it matters whether locks are in the locked or unlocked position, kind of like a binary password).
So one day, it snows in DC and the computer needs to be activated. Unfortunately almost all of the officers (government employees) are on furlough. It's just you and the first officer in fact (you're not a key officer - you're the ice caterer and are considered essential personel). Of course, the first officer's key can do the job of all the other keys, but first someone must crack the secret code and figure out which locks must be turned.
Time is of the essense, our Nation depends upon you.