"Best" column combination

[euklid314]

4-6-8 or 6-8-10 !
Underrated, but really great !

Absolutely! Many players dont know that 4-6-8 has clearly better odds than 5-6-8.

That being said: if you roll, say, 3-3-4-4 as the first roll of a game, do you take 6+8 or the double 7? I tend to the former (since a 7,4,10 would makes a good third column and 5/9 isnt exactly bad either) but I am not decided...

[badboyiain]

Why does 4-6-8 have better odds than 5-6-8 ? I do not understand this.

[euklid314]

It is surprising, but it is a fact.

I want to motivate why it is feasible that 5 has better odds to appear than 4 but in combination with 6+8, the combination 5+6+8 fails more often than the combination 4+6+8.

If you write down all possible 4-dice combinations (1-1-1-1, 1-1-1-2,... ) you will find that many of those combinations that allow to build a 5 will also allow to alternatively build a 6 or an 8. Thus with those dice combinations you will succeed with both 5+6+8 and 4+6+8.

If you look, however, at the dice combinations that include a 4, less of them also allow to build a 6 or 8, thus you will be successful with 4+6+8 but might fail with 5+6+8.

Thus the synergy effects between the numbers 4,5,6,8 matters.

Note, that the combinations are surprisingly close:
6-7-8 succeeds in 92%
4-6-8 succeeds in 91%
5-6-8 succeeds in 90%
2-7-8 succeeds in 89%

all-odd 5-7-9, however, only succeeds in 85%

[Jellby]

Or looking at it from the other side: 6 and 8 are your main targets, in the cases where 6 and 8 fail, 4 is more likely to be an option than 5.

[BarnardsStar]

Or looking at it in yet another way, if we only look at the dice as being odd or even, and remember that two evens make an even but also two odds make an even, then if all four dice are odd or all four dice are even, you can only get even numbers out of that roll. This gives a slight edge to even numbers in general. There are sixteen different ways for the dice to fall if you are only looking at whether each die is odd or if it is even. Only fourteen of them allow you to make an odd number at all, but all sixteen let you make an even number.

This slight edge is obviously not much good if you are comparing the chances of a 2 vs the chances of a 7, but a slight edge is still an edge and it shouldn’t be terribly surprising to see 4-6-8 do (a little bit) better than 5-6-8.

[euklid314]

Or looking at it in yet another way, if we only look at the dice as being odd or even, and remember that two evens make an even but also two odds make an even, then if all four dice are odd or all four dice are even, you can only get even numbers out of that roll. This gives a slight edge to even numbers in general. There are sixteen different ways for the dice to fall if you are only looking at whether each die is odd or if it is even. Only fourteen of them allow you to make an odd number at all, but all sixteen let you make an even number.

This slight edge is obviously not much good if you are comparing the chances of a 2 vs the chances of a 7, but a slight edge is still an edge and it shouldn’t be terribly surprising to see 4-6-8 do (a little bit) better than 5-6-8.

The argument with the odd numbers definitely applies when comparing 5-7-9 with say 4-6-8.

But just looking at a single number (i.e. only one column is open) the probabilities are
2 ... 13%
3 ... 23%
4 ... 36%
5 ... 45%
6 ... 56%
7 ... 64%

That is, 5 is a much better number than 4 - it is clearly more likely to appear. Nevertheless 5-6-8 fails more often than 4-6-8. I would not explain that by odd/even number distribution.

[Arjuna92]

But just looking at a single number (i.e. only one column is open) the probabilities are
2 ... 13% (+13%)
3 ... 23% (+10%)
4 ... 36% (+13%)
5 ... 45% (+9%)
6 ... 56% (+11%)
7 ... 64% (+8%)

That is, 5 is a much better number than 4 - it is clearly more likely to appear. Nevertheless 5-6-8 fails more often than 4-6-8. I would not explain that by odd/even number distribution.

But note that the increase in probability is higher when it's an even number as opposed to an odd one. The column height, on the other hand, increases linearly.

[Ken Corbin]

Having fun with computer skills...
6-7-8 gives you a 29% chance of running the 6 or 8
4-6-8 gives you a 28% chance of running the 4
7-8-12 gives you a 24% chance of running the 12

Now, in real life, you don't target a specific column, you just push the one that has the most progress. I haven't calculated the odds of running any column from a 6-7-8 or 4-6-8 combination, but I would guess it to be close to 50%

[thg0724]

Having fun with computer skills...
6-7-8 gives you a 29% chance of running the 6 or 8
4-6-8 gives you a 28% chance of running the 4
7-8-12 gives you a 24% chance of running the 12

Now, in real life, you don't target a specific column, you just push the one that has the most progress. I haven't calculated the odds of running any column from a 6-7-8 or 4-6-8 combination, but I would guess it to be close to 50%

Did you run 6-7-12? Once you've started with 6-7-12, you need 2 12s, a 12 comes up about once every 7.5 rolls, you're about 20% to complete 15 rolls. But, you're only 12% to complete 15 rolls with 7-8-12. It seems to me that the chances of running the 12 with 6-7-12 should be much higher than with 7-8-12. (It would be great if you could provide the average number of rolls needed for success for the various combinations.)

What was your methodology. Did you start from the lowest spot in each column and with 4-6-8 (and 7-8-12) always choose the 4 (or 12) even when you could advance two spaces on 6-8 (or 7-8)? Similar question for 6-7-8: did you choose a single 6 over a single 7 even if 7 was closer to the top?

[BarnardsStar]

But just looking at a single number (i.e. only one column is open) the probabilities are
2 ... 13%
3 ... 23%
4 ... 36%
5 ... 45%
6 ... 56%
7 ... 64%

That is, 5 is a much better number than 4 - it is clearly more likely to appear. Nevertheless 5-6-8 fails more often than 4-6-8. I would not explain that by odd/even number distribution.

I think I still disagree. 4-6-8 vs 5-6-8 isn’t just about the 4 and the 5. Consider two-column permutations of these four numbers (I won’t bother with 6-8 or 4-5 here):

4-6 . . . 72.1%
4-8 . . . 74.7%
5-6 . . . 73.2%
5-8 . . . 76.9%

Yes, 5 is better than 4 here, but only by a little bit. Adding a possible even column significantly reduces the advantage. Adding two even columns outweighs it entirely, because evens are better than odds.You see the same phenomenon with 4-8-10 (88.3%) vs 4-7-10 (87.6%).

Consider also the “Rule of 28” which calls for adding points for each column (1 for 7, 2 for 6/8 etc up to 6 for 2 or 12) and then adding 2 more if they are all odd, but subtracting 2 if they are all even. Thus 5 gets you one fewer point but breaking the all-even parity loses you two points. (Mind you, I’m well aware this rule is only an approximation; it would tell you 2-4-10 is better than 2-5-10 when it’s not. Still, I think the approximation that “all evens is better than not” holds in general.)