Number of Macro-States w/ 2 6-sided Dice: Formula

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In summary: Love it when a plan comes together. In summary, the formula for calculating the number of macro-states where a micro state is the sum of two dice is n(q-1)+1, where n is the number of dice and q is the number of states the dice can be in. This formula works for any combination of dice and can be used to determine the number of macro-states without listing them all out.
  • #1
Alec Neeson
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If there are 2 6-sided dice. What is the formula used to calculate the number of macro-states where a micro state is the sum of the two dice? I know there are 11 macro-states but I was curious if there was a formula for calculating this.
 
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  • #2
I don't think there is any way to find this out apart from listing all microstates and counting the corresponding macrostates.
 
  • #3
Combine 1, 2, 3, 4, 5, 6 with 1, 2, 3, 4, 5, 6. There are 6 x 6= 36 such combinations:
1+ 1= 2, 1+ 2= 3, 1+ 3= 4, 1+ 4= 5, 1+ 5= 5, 1+ 6= 7.
2+ 1= 3, 2+ 2= 4, 2+ 3= 5, 2+ 4= 6, 2+ 5= 7, 2+ 6= 8. etc.
 
  • #4
HallsofIvy said:
Combine 1, 2, 3, 4, 5, 6 with 1, 2, 3, 4, 5, 6. There are 6 x 6= 36 such combinations:
1+ 1= 2, 1+ 2= 3, 1+ 3= 4, 1+ 4= 5, 1+ 5= 5, 1+ 6= 7.
2+ 1= 3, 2+ 2= 4, 2+ 3= 5, 2+ 4= 6, 2+ 5= 7, 2+ 6= 8. etc.
Can you confirm that I am right in thinking that there can be no way of knowing how many combinations will add to a certain number without actually listing them all (or some variant thereof).
 
  • #5
Alec Neeson said:
If there are 2 6-sided dice. What is the formula used to calculate the number of macro-states where a micro state is the sum of the two dice? I know there are 11 macro-states but I was curious if there was a formula for calculating this.

microstates = q^n, where q = number of states the dice can be in, and n = number of dice

Totally not sure about this, but if we're assuming that dice are always numbered using integers starting at 1 and moving upward sequentially, and label each macrostate with its dice-sum, it seems like it will always be true that the smallest (as it were) macrostate would be n, and the largest macrostate would be q*n. It would also seem true that every integer between n and q*n would also be a legitimately possible macrostate given the integer assumption. If so, the number of possible macrostates in a dice setup could be stated as n(q-1) +1. (edit for the +1)

Disclaimer: Just getting back into thinking and have grown pretty stupid in the last several years.
 
  • #6
Yeah that seems to work! Thank you so much.
 
  • #7
DrClaude said:
Can you confirm that I am right in thinking that there can be no way of knowing how many combinations will add to a certain number without actually listing them all (or some variant thereof).

What makes you think that would be true?

I don't see an easy formula off the top of my head, but I'd be awfully surprised if there wasn't a somewhat elegant way to phrase it.

But maybe you're seeing something that makes you think it's not possible?
 
  • #8
Don't have the notebook in front of me, but I did a 5^3 in the passenger seat and I think it went:

1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1

Somewhat interesting to note, if all 3 sites(n) rolled unique states(q), there are 6 combos; if one duplicate, 3; if triplets, 1. Didn't do a n=4, but if it scales formulaically...
 
  • #9
Alec Neeson said:
Yeah that seems to work! Thank you so much.

Yesssssss
 

1. What is the formula for calculating the number of macro-states with 2 6-sided dice?

The formula for calculating the number of macro-states with 2 6-sided dice is 6^n, where n is the number of dice. In this case, n=2, so the formula becomes 6^2=36.

2. How does the number of macro-states change if we add more dice?

If we add more dice, the number of macro-states increases exponentially. For example, with 3 dice, the formula becomes 6^3=216. This is because each additional dice adds more possible combinations, increasing the total number of macro-states.

3. Can the formula be applied to different types of dice?

Yes, the formula can be applied to different types of dice as long as they have equal number of sides. For example, if we have 2 10-sided dice, the formula would be 10^2=100 macro-states.

4. How is the formula derived?

The formula is derived from the fundamental principle of counting, which states that the number of ways to combine two events is equal to the product of the number of ways each event can occur. In this case, each dice has 6 possible outcomes, so when combined, the formula becomes 6^2=36.

5. Can the formula be used to calculate the number of micro-states?

No, the formula is used to calculate the number of macro-states, which represents the total number of possible outcomes when considering all possible combinations of the dice. Micro-states refer to individual outcomes, so the formula cannot be used to calculate them.

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