The solutions to the puzzles were as follows:
The missing digits were 4, 8, 8, and 2 which respectively encoded as I, N, R, A. Anagrammed to “RAIN”, you needed to look up game #8248 in the BGG database, which was Shoot the Moon, the final answer to the riddle.
This was a fun little one to put together. First I used some Excel coding to find all the games that were composed of 10 unique letters, which would be my first-step solutions. Then some anagram work to come up with appropriate divisors and dividends for each puzzle. Next, more spreadsheet magic to come up with four long division problems that contained all ten digits. Then I had to solve each of the puzzles myself to make sure they were all solvable using only arithmetical logic and arrange them in order of difficulty.
Finally, I wanted a meta-puzzle tying all four together, which luckily wasn’t so hard. All in all I’m pleased with the result. I’m thinking I want to do another variety cryptic for the end of the year, so I better start cracking soon!
Thanks to everyone who entered, or even just tried the puzzles even if you didn’t submit an entry.