In this game, you have an infinite deck of cards. Each time you draw a card it's value is a uniformly distributed integer on the interval [0,C].
The game lasts for R rounds.
You start the game by drawing a card and adding its value to your running total.
At each round you have two choices:
1) draw another card from the deck and add its value to your total
2) add the value of the highest card previously drawn to your total
What strategy, based on the constraints R and C, gives you the optimal total at
the end of the R rounds?
By the way, I was surprised by the result in my solution: If C = 10 and your high card = 9, only draw if there are 45 or more rounds remaining.
But I checked, and it is right.
A randomly drawn card has an expected value of 5, so drawing instead of counting the 9 figures leads to an expected loss of 4 on this round.
But it also has one chance in 11 of drawing a 10, which means you gain 1 point on each of the next 44 rounds. Expected future gain = 44/11 = 4.
So if C = 10 and your high card = 9, draw if more than 45 rounds are remaining, don't draw if less than 45 rounds are remaining, and and if exactly 45 rounds are remaining it is optional, because drawing makes no difference to your expected final score.
Sanity check
