Would it be possible to create a game where one would need to decrypt a simple encryption? This might sound pretty weird, but I was thinking of a game a person could play alone with a pen and paper. 

I got this idea where the person would write a random four letter string on a paper, and would have to decode it. There would be a mean to verify if the person is getting closer to the goal or not. So, I decided to ask here. 

Do you think it would be possible to create a simple game where one would have to decrypt a simple string (a bit like the BitCoin miners do, I guess?), where there would be a logical next step from every point during the decryption and where it would be possible to verify if the contemplated logical next step brings the player closer to the solution or not.

Similar Question: Toy encryption system that provides &quot;hints&quot;

Interesting, but not-quite-it option: Vigenère cipher (Wikipedia article)
Caesar cipher? Its a simple substitution cipher where you can solve with just a pen and paper.

Encryption is done by shifting each plaintext letter, x by n places the alphabet or mathematically, ENCRYPT(x) = (x + n) mod 26. Similarly, the decryption will be the reverse, DECRYPT(x) = (x - n) mod 26. (see wiki http://en.wikipedia.org/wiki/Caesar_cipher#Example)

So, you get a friend to pick a random n and compute the encryption for a randomly chosen four letter string (using free online calculators like http://online-calculators.appspot.com/caesar_ext/). You should not know both n and the plaintext. But the plaintext should be some english word or equivalent in order for it to make sense when you decrypt. 

Then, you start the decryption process by trying out all possible values of n until the decrypted plaintext is make sense. If your first two/three decrypted alphabets is not an valid word, that means your chosen n value is wrong and you should try another n.