When it comes to doing an easy calculation by hand, you are going to run into a problem. An important part of a cryptographic hash function is unpredictability. It is this unpredictability that allows us to use a cryptographic hash function as a proof of work.
For comparison, let's use a simple checksumming method on the alphabet. Normally, a checksum is used to detect copy errors during a transmission or copy of data. A simple method would be to count the number of vowels in the message, and append this number to end. For example:
This message is checksummed
becomes
This message is checksummed|7
A simple checksum like this can be used to check if a vowel as accidentally turned into a consonant, or a consonant accidentally turned into a vowel (not very useful, I know). If either of those two types of errors were to happen during transmission, the checksum wouldn't match on the receiving end, and we'd know there has been an error.
This mrssage is checksummed|7 <--- the checksum is wrong, so there's an error
This simple checksum is also a hash function, as it maps a larger block of data to a smaller chunk of data. Obviously, this is a terrible hash function with an extremely high collision rate, but it's a simple enough example to calculate with pen and paper.
Now let's see what happens when we try to use this hash function as a proof of work. We need to also add a nonce value that can be changed to modify the output of the hash function, without modifying the important part of the message. I'll use the form message|nonce|hash. Let's say that someone performing the proof-of-work needs to find a nonce such that the hash digest is greater than or equal to 10. For example, the following would satisfy the proof-of-work:
This message is checksummed|aaa|10
When the message is hashed (checksummed) together with the nonce, we meet the proof-of-work.
The problem, however, is that our hash function is non-cryptographic, and therefore reversible. Starting with our target value of 10, and our message of This message is checksummed, we can perform the checksum on the message and then do simple arithmetic to generate an acceptable nonce. There is no unpredictability in the system.
If you use some sort of pen and paper hash function, you are almost certainly going to run into this problem. Likely, those attending your class will see right through it, and not understand the purpose of the proof-of-work. In the case of this example, it would be pointless.
If you choose to use a method like this, it is probably worth explaining to your class the significance of cryptographic hashing functions. Otherwise the entire concept of proof-of-work doesn't make sense.
