James Ellis at GCHQ came up with a pen-and-pencil example of public key cryptography, something he called "non-private encryption." He intended it as an illustration of a public key system.
The gist of his system is something like this: You start by creating a lot of short riddles, each having a unique solution, which could for practical purposes be mathematical equations. For each of these riddles you create an encryption key. The intention is that the solution to the riddle is the ID of the key. The full list of (riddle [ie. key ID] / encryption key) is posted somewhere public.
When someone wants to send you a secret message, they pick a pair at random. They solve the riddle, to get the key ID. They can then encrypt the message they only want you to see, by using the key they chose from the list. So that you can decrypt the message, they note the key ID somewhere in cleartext in the letter head.
The secrecy of this system depends on the number of riddles, and the time it takes to solve each one for the key ID (an attacker with the all key IDs can simply look up the key). For instance, if you have 10,000 pairs, and each riddle takes a day to solve, it should last about 30 years against a single attacker.