Generating Splendor Card Values for an Arbitrary Set of Gems

Context: Got bored trying to map card values in a spreadsheet for a Pokemon re-theme of Splendor, and instead nerd sniped myself into generating them with formulas/algorithms/heuristics.

Then I nerd sniped myself again, this time wondering if the card values can be generalized to any number of gems than just 5, and if that kept the balance of the cards (gameplay overall would need a lot of tweaks I’m sure but that’s a bridge for later). The Pokemon re-theme would benefit greatly from it.

A lot of this was posted and discussed in the ReRoll WhatsApp group. I’m collecting and re-posting things verbatim here so it’s accessible to others outside the group too. Might edit things later on so we could have an accessible article or read out of it.

A big thanks and shout-out to @nemo for doing a lot of heavy-lifting and research on this.

This was the raw discussion pertinent to what was discussed. I’ve snipped out some of the stuff that was ancillary to this forum post’s topic, or was incomplete (like the image of my notes – will add it in full later when I’m done with it).

[18/05, 23:41] @mohdjas: Hi everyone. If anyone knows or has come across a heuristic/algorithm to generate Splendor card values (gems to pay to buy and VP), could you please share a link or point me to where I could find it? Thanks!

[18/05, 23:43] @nemo: Curious about what’s the need for this.

[18/05, 23:48] @mohdjas: Started off as being lazy and bored trying to map card values in a spreadsheet for a retheme, and instead generating them with a bunch of formulas.

Now I’m wondering if the card values can be generalized to any number of gems than just 5, and if that keeps the balance of the cards (gameplay overall would need a lot of tweaks I’m sure but that’s a bridge for later).

[18/05, 23:48] @mohdjas: I went through your boardgame papers repo, couldn’t find anything about Splendor in it.

[18/05, 23:50] @nemo: Hmm. I’ll try to play around to find some answers for you. Worth checking BGG for any variants that play with the balance as well.

[18/05, 23:52] @nemo: Also a good question for Boardgames StackExchange.

[18/05, 23:54] @mohdjas: Thanks, appreciate the help! I’ll look through the variants. Dunno if Marvel Splendor made any changes, or even Pokemon Splendor (which incidentally was the one that made me go for a retheme/rework). But hopefully can eke something out of them.

[19/05, 11:34] @nemo: The Time-to-Point Ratio in Splendor calls it a five-point crystalline distribution.

Plus, I found a spreadsheet at Splendor cards list - Google Sheets

[19/05, 11:43] @mohdjas: Wow, this is great! Thanks so much! I found a BGG post where someone had asked about adding two new gems but it wasn’t a very long or productive discussion, at least to card point structure: Pokemon Splendor - How Do You Think Adding a New gem color or 2 will affect the game? | Splendor

Looks like I have a lot of reading to do, thanks again!

[19/05, 13:38] @nemo: The rules for noble cards are simple. two pairwise sequences cover all of them where a-e are the 5 colors:

  1. 4,4 for each of the sequential pairs (ab, bc, cd, de, ea)
  2. 3,3,3 for each of the 3-sequence pairs (abc, bcd, cde, dea).

After an hour of reading through Multiset - Wikipedia lots of times, I don’t think there is a simple generator that was used. There’s just too many combinations unused. For example, 6 gem cards only use 6,0,0,0,0 and no other combination (such as 5,1 / 4,2,2 / 4,1,1,1 / 3,3 / 3,2,1 etc). The highest count used happens at 10 where the only combination is (7,3), and not surprisingly at 12 or 14, which actually use 6 and 5 as the highest count.

So this is likely just balancing and “game-design”.

[19/05, 13:39] @nemo: There are some inelegant ways of generating the list, but I don’t think you’re looking for those.

[19/05, 14:35] @nemo: You can tweak the balance quite simply by either changing the combination used for a gem count (6 0 to 4 2), or just adding it as an extra card.

[19/05, 15:00] @mohdjas: I’ll look into the points-to-turns ratio that the first blogpost mentioned and see if I can use that somehow.

One of my lines of thought is that we might be looking into generating something logically when it most likely was that the VP were added as a result of game balancing and design. After all, if each Tier 1 card had atleast 1VP, one would just increase the number of VP on other tiers in addition to the game-end trigger.

[19/05, 15:02] @mohdjas: First though I’m going to try and see how to make a 5-gem formula, and then try to generalise from there. Easier to tackle it that way and at least verify with the existing game.

Reserved for future use.

Reserved for that not in the present nor the past.

The nobles are generated in two pairwise sequences:

  1. 4,4 for each of the sequential pairs (ab, bc, cd, de, ea)
  2. 3,3,3 for each of the 3-sequence pairs (abc, bcd, cde, dea).

This makes for a nice symmetric decision. However, the Gem cards are much more hand-picked.

I tried to convert the existing list of cards into a single list, and found Prime Factorization as the simplest representation. The gem cards can be found doing a prime factorization of the following list:

64,243,343,441,539,625,1155,2475,5775,18634,33075,37044,1452605,19487171,22370117,155897368,4861163384,75499144875

The colors are denoted by the prime numbers 2,3,5,7,11, and their coefficients denote the count for each card.

As an example, 243 factorizes to 35, so there is one card that has the “3-color-gem”*5 as the cost.

37044 factorizes to 223373, so there is a card with costs as (2 3 3).

Each of the colors have a symmetric set of cards, so you only need one list.