Maximum number of possible synergies (+ some comps)

Hey, I wanted to take a more theoretical approach to this game. A content creator that I mod for tries a lot of crazy stuff (some people call him the German Vandiril, our full Sylas Ult Interaction Video was up before his video) and he also tried a lot of comps with many champions. 9 champions and 21 trait points total is actually possible in most games, for more champions you need a Force of Nature.

#The Question

Now I asked myself what the highest possible number of synergies you can achieve is. I'll be excluding Spatula-items in this calculation because you'll be building Force of Nature_s_ with them, but other than that, it's pretty straightforward. We want to have the highest number of the sum all synergy points, even if it does not achieve something. E.g. Knight 3 counts as 3 points, although it works like Knight 2. For optimal comps, that means all synergies must be active. You can't have stuff like Glacial 1 trailing behind (except for Ninja 1 / Exiled 1 which is active at that point), or you could pick any champions to achieve the maximum number of points, giving up all (or just most) synergies in exchange.

#The Answer

In theory this is very easy again. Pick every 3-Trait-Champion you can, and then pick as many "normal" champions as you can. If n is the number of champions, (and let's say n >= 7 for all lategame builds), then the maximum possible number of synergies is 4*3 + (n-4)*2. Simplified it's 2n + 4. Keep in mind that this is a hypothetical number for small n, because you can't find a comp that achieves this level of synergy.

** Conjecture: For n >= 12, there exists a comp with n champions and 2n + 4 synergies. **

#The Comps

I only will give the comps for n = 12, 13 and 14. I don't think it is possible to find a comp with the constraints I outlined for n = 9 to 11, and n = 8 and below is boring.

You obviously need to pick all the 3-Trait-Champions. For n = 12, your team is This team gives you Pirate, Blademaster, Gunslinger, Ninja, Elementalist, Yordle, Shapeshifter, Demon, Imperial, Wild, Glacial. All of these are on the first level except for Gunslinger 4 (second level).

For n = 13, take the above team and add or . That does not add extra synergies, but more synergy points as outlined above. You'll have 3 Demons and 4 Shapeshifters / Elementalists, but it still achieves the 30 synergies that I demanded.

For n = 14, you could add both Brand and Elise. That's boring, and I found another team that satisfies my calculated 32 traits:

You could swap Lulu with Veigar, or add Veiger for n = 15. This team gives you Demon, Dragon, Glacial, Blademaster, Elementalist, Imperial, Ninja, Pirate, Gunslinger, Wild, Shapeshifter and Yordle on their first level, as well as 4 Sorcerers (also first level, but one too many actually).

For even higher n, construction is easy because you can add champions that have two traits already present. Veigar, Graves, Nidalee, Elise, Brand or Yasuo. Yasuo is a special case because Blademaster is already there, but Exiled isn't. However Exiled works on its own so it's fine.

Adding all of the above to the 14 champion team gives you the perfect team for 20 champions with 44 synergy points. For 21, you'd need to find a different team to work with, but after that you can add champions with traits already present. This is why I believe for every n >= 12, there exists a comp that satisfies 2n + 4 synergies.