I watched a video awhile ago from Ted-Ed called the "cheating royal riddle": https://www.youtube.com/watch?v=hk9c7sJ08Bg. The riddle is as follows:
- You're the advisor in a competition where four contestants roll both dice 20 times, in private, and add up the results.
- The red die has the numbers
2, 7, 7, 12, 12, and 17on the six sides, and the blue die has
3, 8, 8, 13, 13, and 18. The dice are fair.
- A contestant should be disqualified by you, the advisor, if you're at least 90% sure that the score they reported wasn't actually their total.
- The highest-scoring player who wasn't disqualified is the winner.
In the video, the contestants
D, reported values
B was disqualified for not being possible (too high).
C was disqualified or being improbable (would require rolling highest numbers on all 20 rolls), and
D was disqualified for not being possible (not a multiple of
A was the only one left not disqualified, so she won.
What would the optimal strategy look like for this game if you were one of the contestants and wanted to cheat and not be detected by the advisor?
My first attempt is to honestly play the game mostly, but have a "cheat" round every 3 rounds, where I take the lowest number rolled and manually flip it to the highest number on that die. But I cannot figure out how to calculate the odds of that strategy on average.
I guess there is perhaps a calculable number that you can just calculate and report, but it would be interesting to also know if there is an optimal strategy to "play the game out" and be very likely to score higher than your opponents but not too high as to exceed the advisor's 90% threshold for cheating; that strategy would work even if you were observed by a proctor. I guess if you "play the game out", there is always a possibility that you get extremely lucky and are disqualified for cheating even if you didn't, so any strategy to improve your odds will make that situation more likely, so perhaps the best strategy is to just calculate the number and report it to ensure you're right below the threshold of being disqualified.
I'm working on a simulator for this just out of curiosity, but I figured I would ask here. I am new to this SE (normally on SO), but "Solving mathematical puzzles" showed up on the on-topic help (https://math.stackexchange.com/help/on-topic) for this SE, so I hope this is an appropriate place to put this question.