How Can You Roll a 13 Using 2 Dice?

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Discussion Overview

The discussion revolves around the question of how to achieve a total of 13 using two dice. Participants explore various mathematical interpretations, alternative numbering systems, and creative labeling of dice, with a focus on theoretical and conceptual approaches rather than practical applications.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests using a base 5 numbering system for one die, proposing that rolls (5,3) and (6,2) could yield 13, but acknowledges the complexity of expressing this idea.
  • Another participant counters that the addition of values from different bases does not yield 13, suggesting instead to replace the standard die values with higher numbers to achieve the desired total.
  • A different approach involves labeling one die with values that allow for a broader range of sums, such as 0, 6, 12, 18, 24, and 30, to achieve totals up to 36.
  • Some participants propose labeling the dice in a way that allows for combinations that sum to 13, such as using a straight run from 1 to 6 and then 7 to 12.
  • One participant emphasizes the desire to keep the traditional die faces while exploring alternative mathematical interpretations, such as using different bases.
  • Several participants discuss the idea of interpreting the results of the dice in various bases, such as base 8 or base 12, to achieve a total of 13, with some expressing that this feels like "cheating."
  • Another suggestion involves rolling one die at a time to clarify which die contributes to a higher base value, allowing for the possibility of achieving 13 through unconventional means.

Areas of Agreement / Disagreement

Participants express a range of views on how to achieve a total of 13 with two dice, with no consensus reached. Some agree on the potential of using alternative bases or labeling, while others challenge the validity of these approaches.

Contextual Notes

The discussion includes various assumptions about the nature of dice and their values, as well as the implications of using different bases, which may not be universally accepted or understood.

dkotschessaa
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Originally posted on another forum with the title "are some things impossible?" The question was more of a philosophical one, but it got me thinking mathematically.

I thought I had a solution like this:

Put die #1 in a base 5 numbering system, you get the following values for the 6 rolls for die 1.
Roll Value (in base 5)
1 1
2 2
3 3
4 4
5 10
6 11

If you keep the second die in base 10 then, the rolls (5,3) and (6,2) give you 13.

But that's like trying to add 1 (binary) to decimal 2 and getting 3. I think I'm onto something but I don't know how to express it yet. Any ideas?

-Dave KA
 
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No, [itex]11_5+ 3_{10}\ne 13[/itex] in any base! If you are determined to get a roll of 13 with two dice, replace 1, 2, 3, 4, 5, 6 on one die with 2, 4, 6, 8, 10, 12. then "13" would correspond to a roll of (1, 12).
 
Last edited by a moderator:
You could label one die with 0,6,12,18,24,30.
Then with the other, normal die, each result in the range 1-36 will be equally likely.
 
If one desired to make 13, in essence, the new 7, then you could label the dice as a straight run from 1 --> 6, and then 7 --> 12, in which case the least likely rolls would be 8 and 18.

6, 7 --> 13
5, 8 --> 13
4, 9 --> 13
3, 10 --> 13
2, 11 --> 13
1, 12 --> 13

6, 12 --> 18
1, 7 --> 8
 
I think when the OP (on the other forum) said "ordinary dice" he meant those labeled 1-6, meaning one dot up to six dots. So I was trying to come up with a way to do it without changing the dots themselves. That's why I thought of a different base numbering system. So the symbols could stay the same but represent something else.
 
dkotschessaa said:
I was trying to come up with a way to do it without changing the dots themselves.

Not sure if this is what you were getting at, but here are some equivalencies to base 10 dice rolls in other bases...

10 --> b4, b5, b6, b7, b8, b9
---------------------------------
02 --> 02, 02, 02, 02, 02, 02
03 --> 03, 03, 03, 03, 03, 03
04 --> 10, 04, 04, 04, 04, 04
05 --> 11, 10, 05, 05, 05, 05
06 --> 12, 11, 10, 06, 06, 06
07 --> 13, 12, 11, 10, 07, 07
08 --> 20, 13, 12, 11, 10, 08
09 --> 21, 14, 13, 12, 11, 10
10 --> 22, 20, 14, 13, 12, 11
11 --> 23, 21, 15, 14, 13, 12
12 --> 30, 22, 20, 15, 14, 13

If, as per my previous post, one wanted to make 13 the new 7 (i.e. the most likely roll), base 4 would be the way to go.
 
Last edited:
dkotschessaa said:
Originally posted on another forum with the title "are some things impossible?" The question was more of a philosophical one, but it got me thinking mathematically.

I thought I had a solution like this:

Put die #1 in a base 5 numbering system, you get the following values for the 6 rolls for die 1.
Roll Value (in base 5)
1 1
2 2
3 3
4 4
5 10
6 11

If you keep the second die in base 10 then, the rolls (5,3) and (6,2) give you 13.

But that's like trying to add 1 (binary) to decimal 2 and getting 3. I think I'm onto something but I don't know how to express it yet. Any ideas?

-Dave KA

Use dice of different colors. Choose one color and add 6 to each face.
 
Perhaps one easy way is to interpret the dice result as in base 8; thus 6 + 5 = 13 (base 8).

It sounds like cheating to me, anyway. :)
 
Dodo said:
Perhaps one easy way is to interpret the dice result as in base 8; thus 6 + 5 = 13 (base 8).

I think that's what I'm after. Simple! I'm not good with different bases.

It sounds like cheating to me, anyway. :)

Totally. It was just a mathematical answer to a non-mathematical question on a board of sci-fi geeks.

-M
 
  • #10
Dodo said:
Perhaps one easy way is to interpret the dice result as in base 8; thus 6 + 5 = 13 (base 8).

It sounds like cheating to me, anyway. :)

Consider the base as 12 and roll a 1 and 1. If one objects that you can't tell which die is the 12's digit one can roll one die at a time so you know which is the first of the ordered pair. No one said that there can't be gaps in the result rolled. But what I think is the best answer (in line with the op's original post ) is to consider the result in base 5; then, if you roll a 6+2, it would be 13 in base 5. Dies don't have bases, you just count the number of dots, 8 is 13 in base 5. You could also roll a 6+4 to get a 13 in base 7
 
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