1-4-24 Just for Fun

[Crimson]

Today, I engaged in a game called 1-4-24 with a friend of mine. There are 6 dice, and you roll, removing at least once die per role, or more if you like. To "qualify," you must remove a 1, a 4, and the remaining dice have to = as close to 24 as possible. For example, if I rolled a 2, 6, 5, 1, 3, 4, on my first roll, I would remove the 1, the 4, and the 6, and possibly the 5, because its a pretty solid number. There would be two die left, I roll and get a 4 and a 2, I take the 4 obviously, and the last roll I get a 3, giving me a final score of 6+5+4+3=18.

Now it's his turn, and he trys to out do me.

The question is this, what do I do if on my first roll I roll all 6's, and no 1's or 4's? I have to take at least one, how many do i take and how many do i keep rolling?

Of corse, position matters here, as if he ended up with a 12, it wouldnt really matter as much....

Was wondering what all you math wizzes could do with EV and Odds with this one, he keeps wanting to play for money, and with a solid strategy, I'd say what the hell....

Thanks : )

[bobman0330]

Clarification: If you don't pick up a 1 and a 4, you lose?

[Crimson]

Yes, if you dont take a 1 and a 4, then you lose, so Ideally, your "hand" is a 1, a 4, and 4 sixes.

[bobman0330]

My thoughts, part 1:
If you take 4 6s, you will roll perfect on the next roll 1/18 times. You will bust out 4/9 of the time. The remaining half of the time, you make your hand 1/6 of the time. You make a hand, overall, 14% of the time (1/12 + 1/18).

If you take 3 6s, you have 3 dice to make a 4 and a 1.
will roll neither 30% of the time. You will roll a 4 and a 1 8% of the time. The remaining 62% of the time you will roll either a 4 or a 1, but not both.

So, 8% of the time you make a hand worth, on average, 22.25 (rolling one die twice yields an EV of 4.25, I think [1/6 * 6 + 1/6 * 5 + 1/6 * 4 + 1/2 * 3.5]).

62% of the time, you take your 4 or 1 and reroll. On your first roll, you will catch the other one 31% (cumulative: 19%). This will give you a hand worth 22.25 again. The remaining 69% of the time, you miss. You will hit on the last roll 1/6 = 17% of the time, giving you a hand worth 22.25. (cumulative = 7%).

If you miss both, you take the highest valued die and roll more times as about a 30-1 dog.

So, if you take 3 6s, you make a hand about 35% of the time, valued at 22.25.

It's very possible some of the math in the 3 6s section is off. I'll look at this some more later.