What is the fastest possible game in Splendor?

Question

In the game of Splendor, what is the fewest number of turns you could take to get 15 points?

Answer 1

I have found 15 points in 15 turns, which works even for 3 player rules:

1. Take: white, blue, green
2. Take: white, blue, green
3. Take: white, blue, red
4. Claim: white card (3 blue)
5. Take: green, red, black
6. Claim: white card (2 red, 1 black)
7. Take: white, blue, green
8. Claim: white card (4 green) +1 pt
9. Claim: white card (6 white) +3 pt
10. Take: 2 white
11. Claim: blue card (7 white) +4 pt
12. Take: white, blue, black
13. Take: 2 white
14. Claim: blue card (7 white, 3 blue) +5 pt
15. Claim: green card (4 white, 2 blue, 1 black) +2 pt

Although, I don't have proof that 14 turns game is not possible.

Answer 2

Update: here used to be my solution for 16 points in 16 turns, but since then I've managed to find an even better solution with my bruteforce tool. Still 15 turns only, but this feels so efficient that I doubt a 14 turn one exists. This also exploits the green/blue imbalance that I mentioned below.

16 points in 15 turns:

 1. Take    blue,  red, ⚫ black
 2. Take    blue,  red, ⚫ black
 3. Take    blue,  red, ⚫ black
 4. Claim   blue card  (3 ⚫ black) + 0 pt
 5. Take    blue,  green,  red
 6. Claim   blue card  (4  red) + 1 pt
 7. Claim   blue card  (6  blue) + 3 pt
 8. Take    blue,  green,  red
 9. Take    blue,  green,  red
10. Take    blue,  green,  red
11. Claim   blue card  (2  blue, 2  green, 3  red) + 1 pt
12. Claim   blue card  (5  blue) + 2 pt
13. Claim   green card (7  blue) + 4 pt
14. Take    blue,  blue
15. Claim   green card (7  blue, 3  green) + 5 pt

One thing I've noticed is that there's an interesting disbalance among the 2 pt cards for 5+3 gems: only 2/5 cards (blue and green) give the bonus of the same color as one of the requirements. Feels like something that might yield an even better solution.

Answer 3

EDIT: fixed violation of limit on taking 2 of same gems (Forbidden when <4 gems remaining).

First of all, @Arcanist Lupus has provided good answer with simple yet efficient approach and also provided link to cards table in the comments, so I want to thank them. But looking at the table I noticed that 4 pts. cards have better conversion rates than 5 pts. ones, and a lot of gems are lost when going straight for 5 pts. cards.

So, assuming all required cards are available, no nobles and no interference from other players, here is my solution:

Cards I'm going for: White 4 for 7 blacks; Blue 4 for 7 whites; Green 4 for 7 blues; Green 3 for 6 greens.

Steps:

---

1. +2 black
2. +2 black
3. +1 black, 1 green, 1 {any}
4. +1 black, 2 {any}
5. +1 black, 1 white, 1 green. Return 3 {any}
6. Claim White 4 card for 7 blacks. (1 white and 2 green remain in the stash)
7. +2 white
8. +2 white
9. +1 white, 1 blue, 1 green
10. Claim Blue 4 card for 6 whites + card. (1 blue and 3 green remain in the stash)
11. +2 blue
12. +2 blue
13. +1 blue, 1 green, 1 {any}. Return 1 {any}
14. Claim Green 4 card for 6 blues + card. (4 green remain in the stash)
15. +1 green, 2 {any}
16. Claim Green 3 card for 5 greens + card

Done. 15 points in 16 turns.

• steps 3 and 4 are just about getting 2 blacks, since it's forbidden to get 2 at once at that point. Alternatively you can take golden tokens and secure couple of cards for yourself, in that case in step 15 just take 2 greens (you will have 1 less, so it would be legal move).

Answer 4

The top answer by Marcin_smu does not work in two player games. On move 10, he picks up 2 white chips but has one in his hand, in 2 player this is not possible. In 3 player or 4 player, it would be possible.

I have 15 points in 15 turns for 2 player:

1. 1 blue + 1 black + 1 red (0 points, 1u1b1r, 0 cards)
2. 1 blue + 1 black + 1 red (0 points, 2u2b2r, 0 cards)
3. 1 blue + 1 black + 1 red (0 points, 3u3b3r, 0 cards)
4. Claim the 3 black for a blue card (0 points, 3u3r, 1 blue)
5. 1 blue + 1 green + 1 red (0 points, 4u1g4r, 1 blue)
6. Claim the 4 red for a blue card plus 1 point (1 points, 4u1g, 2 blue)
7. Claim the 6 blue for a blue card plus 3 points card (4 points, 1g, 3 blue)
8. 1 blue + 1 blue (4 points, 2u1g, 3 blue)
9. Claim the 5 blue for a blue card plus 2 points card (6 points, 1g, 4 blue)
10. 1 blue + 1 blue (6 points, 2u1g, 4 blue)
11. 1 blue + 1 green + 1 white (6 points, 3u1g1w, 4 blue)
12. Claim 7 blue for a green card plus 4 points (10 points, 1g1w, 4 blue 1 green)
13. 1 blue + 1 blue (10 points, 2u1g1w, 4 blue 1 green)
14. 1 blue + 1 green + 1 white (10 points, 3u2g2w, 4 blue 1 green)
15. Claim 7 blue 3 green for a green card plus 5 points (15 points, 2w, 4 blue 2 green)

note: a duplicate version of this could be done with green but not red, black or white because the 2 point 5 chip card is not monochrome

Answer 5

Each color has a single card worth 5 points, which can be taken by 7 gems of one color and 3 of another. Conveniently, the maximum number of gems you can hold at once is 10, and in a 4 player game there are 7 gems of each color available, which means that you can go after these cards immediately.

For example:

1. 2 red
2. 2 red
3. 1 red, 1 black, 1 {any}
4. 1 red, 1 black, 1 {any}
5. 1 red, 1 black, 1 {any}, return 3 {any}
6. Claim the 5 pt black card
7. 2 black
8. 2 black
9. 1 black, 1 white, 1 {any}
10. 1 black, 1 white, 1 {any}
11. 1 white, 1 blue, 1 {any}, return 3 {any}
12. Claim the 5 pt white card (keep the blue)
13. 2 blue
14. 2 blue
15. 1 blue, 1 white, 1 {any}
16. 1 blue, 1 white, 1 {any}
17. Claim the 5 pt blue card.

Congratulations! You've won!

It's possible that there's a faster Nobles strategy, but I'm not sure. 3 Nobles requires at least 12 turns of taking cards, which means you have a total of 4 spare turns for gem collecting (5 to tie my score, 4 to win outright), and it takes three gem turns to get your first 3 cards. Plus you have to stay in cards that produce only 3 different colors.

Answer 6

As a partial solution, here are some of the qualities I suspect will need to be part of such a "perfect game":

1. All Noble tiles claimed (since they represent a large source of points for no additional action taken).

2. Very few turns where chips are taken, and a turning point somewhere in the middle of the game after which no chips are taken (since every turn taking chips is a turn you aren't taking cards which gain you points).

Additionally, I suspect that the structure of the game will be something like (assuming a 2-player setup, so there are 9 points available in nobles):

1. Spend a few turns taking chips and acquiring cheap level 1 cards.

2. Get enough level 1 cards that all future purchases are free, probably earning the first noble near the end of this step.

3. Get 1 or 2 level 2 cards worth 2-3 points, earning the second noble.

4. Get a level 3 card and the third noble.

It might be possible to shortcut step 3 if you get a couple of the 1 point level 1 cards in step 2, but I don't think that there's a lot to be gained going that way (the 1 point cards are not particularly cost-effective).