What's the Best Strategy for Scoring High in a Six-Dice Game?

[Graff]

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In this game you roll six dice. After each roll you must take at least one die but you can take more than one as well. After you take a die you have the ability to roll again. In order to score you must have the qualifying die rolls of (4) and (1). The four other spots compromise your scoring dice. Ties are effectively losses and the highest score wins.

In this scenario you have the ability to roll, which means that you can take any number of dice before rolling again. Do you take the (5) before rolling again? Math would be awesome.

I have a vague goal of finding a "perfect" strategy for this kind of game, is this possible?

 

[Simon Bridge]

You have to break it down.
i.e. in order to have a scoring hand, you need two of the dice to read a one and a four ... there must be at least one of each.

The rest depends on the scoring - you want high numbers so this suggests you want to reroll 1 and 2, but not 5 and 6. Though the details depend on your opponent's hands. (Depending on if you want to win or just not come last.) You can figure out what the odds of beating it are from different starting points.

 

[Graff]

You have to break it down.
i.e. in order to have a scoring hand, you need two of the dice to read a one and a four ... there must be at least one of each.

The rest depends on the scoring - you want high numbers so this suggests you want to reroll 1 and 2, but not 5 and 6. Though the details depend on your opponent's hands. (Depending on if you want to win or just not come last.) You can figure out what the odds of beating it are from different starting points.

I'm just talking about the image I posted, I don't know how to figure out the odds form the starting point of my image.

 

[Simon Bridge]

And I just told you how to start, and some of the issues to think through.
Though this is easier than trying to figure a general strategy for the whole game :)

Break it down to the options - you can reroll the 1, the 5 or both of them.
So work out the odds of improving your situation in each case.
What are the likely scores for your opponents?