I want to make a game where there's a total of 10 general weapon types. I want any given character to be able to be proficient with 2 of the weapon types. Looking at the fact that 10 is an even number, a perfect rock-paper-scissors relationship wouldn't be possible. "C(10,2)=45" however is an odd number, which should be workable into a rock-paper-scissors system.
Since the final number of total weapon pairings is an odd number, I reckon there should be an intuitive and systemically organized way to get a rock-paper-scissors relationship to exist where each weapon pairing is at an advantage against 22 of the other pairings while being at a disadvantage against the remaining 22 pairings. Assuming the 10 weapon types that a character can be proficient with 2 of are labeled A through to J, how should I organize it?
P.S.: A smaller example that runs into the same situation would be "C(6,2)=15"; weapons labeled A through F; if that's more convenient for explanation purposes. Please let me know if this question belongs under different tags or even over on the Mathematics section of SE instead of here; I wasn't sure where to put this question.
Answer
The simplest way to do it would be to lay out your weapon combinations in whatever order you see fit in a circular list. Each combination could have a bonus against the 22 combinations after it in the list, and a weakness against the 22 before it.
The player might appreciate a more straight-forward or easier-to-recognise logic to that order -- some sort of pattern.
With your 6 weapon example, you could have:
1. AB AC AF
2. BC BD BF
3. CD CE CF
4. DE DA DF
5. EA EB EF
Here you have a whole lot of rock-paper-scissors. Every combination of weapons has a "primary" and a "secondary" weapon according to a rock-paper-scissors between the first 5 weapon types, with the 6th weapon type (F) always being secondary (since, as you pointed out, we need an odd number for rock-paper-scissors).
We now have 5 super groups, and can have a simple 5-way rock-paper-scissors between them (group 1 beats groups 2 and 3, 2 beats 3 and 4, and so on).
If two characters are using the same primary weapon, we have a 3-way rock-paper-scissors. The way I've laid it out here, you could have AB beat AC, AC beat AF, and AF beat AB.
Expanding this to 10 weapons, you can divide the 45 combinations into 9 primary weapon groups of 5 combinations, with the 10th weapon always secondary:
1. AB AC AD AE AJ
2. BC BD BE BF BJ
3. CD CE CF CG CJ
4. DE DF DG DH DJ
5. EF EG EH EI EJ
6. FG FH FI FA FJ
7. GH GI GA GB GJ
8. HI HA HB HC HJ
9. IA IB IC ID IJ
While this kind of pattern may not be super easy to memorise, a well-designed UX could do a lot to make it clear which weapon is a character's primary weapon, which primary weapons beat which other primary weapons, and which secondary weapons beat which other secondary weapons (conveniently in the same order as primary weapons, except for the always-secondary weapon J).
I hope that makes sense!
With 11 weapon types, with 55 combinations, which, as you've pointed out, is divisible by the number of weapons, you don't need a special "secondary only" weapon:
01. AB AC AD AE AF
02. BC BD BE BF BG
03. CD CE CF CG CH
04. DE DF DG DH DI
05. EF EG EH EI EJ
06. FG FH FI FJ FK
07. GH GI GJ GK GA
08. HI HJ HK HA HB
09. IJ IK IA IB IC
10. JK JA JB JC JD
11. KA KB KC KD KE
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