Hard endgame puzzles

[florrat]

The endgame in hanabi is complicated. This post is not a guide about endgame strategies, but instead a list of puzzles with very complicated solutions. I think even the best hanabi players will have a hard time with these.

For every puzzle I will link to the board if I have it, but I will also give all the essential information needed for the puzzle. In some puzzles, it depends on the deck what the best move is, but since you don't know the deck, your goal is to maximize the expected score. Describe for each puzzle what the best move is, and any later decisions which are not trivial.

Puzzle 1: #37853032, florrat's turn with 4 cards in deck.
2 players, 4 cards in deck, 3 clues. To play: G3 and up, Y5, W5. 1 copy of G3 and G4 are in discard.
Your hand has only useless cards. Partner's hand: R3 R1 B2 G5 W5, the G5 and W5 are fully clued.



Puzzle 2: (no seed)
2 players, 3 cards in deck, 1 clue. To play: M4 and up, R5, Y5. 1 copy of M4 is in discard.
Your hand: nothing clued, recently you were incredibly low on clues, so you might have M4 in hand.
Partner's hand: B1 Y5 R5 B1 M5. The M5 is fully clued, no other card is clued

Puzzle 3: #35314150, gamblingoof's turn with 3 cards in deck.
2 players, 3 cards in deck, 3 clues. To play Y5, M3 and up. Multicolor cards in discard: M3, M1, M2.
Your hand: 2 of them are marked as multicolor, but without number information, so one of them might be useless. They were first clued as part of a play clue on the Y3 in the endgame. They are not M3s, because partner would have told you that.
Partner's hand: W3 R3 R2 Y2 Y5 M4 with the Y5 and M4 fully clued.

[Romain672]

#1 :
I don't get the problem... Both discard or clue are fine. You need to discard one time, not Notafool, but that's all. No matter when you discard.


#2 :
You can get max score 75% of the time I think (or 62,5%?) if you think the same than the other player (except if 4m is last). There is only one move off, but it's enough to give all informations.

My personnal opinion :
You give 1 and assume he understand both 5 (else you could just empty clue).
If you have 4m in :
- second position : he play 5y
- forth position :he discard his second 1b
- fifth position : he discard his first 1b
- first, third, or nowhere : he play 5r


#3 :
Same, I don't see what you want here... You clue green. He play 5y. You clue green again. No matter what happen, it will be fine.

[florrat]

#1 :
I don't get the problem... Both discard or clue are fine. You need to discard one time, not Notafool, but that's all. No matter when you discard.

It does matter. The question is: what is the best moment to discard? Immediately (risking that you don't draw anything playable) or later (risking that Notafool draws G4)? One of them leads to a higher probability of winning.

#2 :
You can get max score 75% of the time I think (or 62,5%?) if you think the same than the other player (except if 4m is last). There is only one move off, but it's enough to give all informations.

My personnal opinion :
You give 1 and assume he understand both 5 (else you could just empty clue).
If you have 4m in :
- second position : he play 5y
- forth position :he discard his second 1b
- fifth position : he discard his first 1b
- first, third, or nowhere : he play 5r

Okay, you have to know your partner really well to have such complicated rules (I would not play 5th position blindly if you discard first 1b ). And if your partner plays 5r, I assume he then does something similar next turn, to let you know exactly where your M4 is? I guess that works if you know your partner really well. There is a solution without any nonstandard meaning of discard clues.

#3 :
Same, I don't see what you want here... You clue green. He play 5y. You clue green again. No matter what happen, it will be fine.

That's mostly right, but you can do a little better than that.

[Romain672]

#1 : All the possibility who can give 29 (removed 4g and 3g in last, and xx xx 3g 5y) :
(1) : xx 3g 4g 5y
(2) : 3g xx 4g 5y
(3) : 3g 4g xx 5y
(4) : 3g 4g 5y xx
(5) : 3g 5y 4g xx
(6) : 4g 3g xx 5y
(7) : 4g 3g 5y xx
(8) : 4g 5y 3g xx
(9) : 5y 3g 4g xx
(10) : 5y 4g 3g xx
I still don't get the difference. Perhaps one time you need to play your first card 'in blind' if you clue first, but that's all...
Can you tell me the possibility where you doesn't do 29 (guess if you clue at this turn : I'm pretty sure it's less good) ?



And for #2 you still get max score with that ?

[florrat]

#1:
Suppose we're in possibility 1. If you waste a clue, you get 29, if you discard you get 28.
Suppose we're in possibility 8. If you waste a clue, you get 28, if you discard you get 29.
There are other possibilities where there is a difference. So the question is: which one is better considering all 10 possibilities?

#2: yes, whenever possible (so whenever m4 is not bottom) (I am assuming that partner can clue my m4 with a single clue if I have that card, which seems reasonable enough).

[fuzzycoatimundi]

1. [strike]Your partner/Notafool should draw the penultimate card. Hence, you/florrat should discard/play to draw the fourth-to-last and third-to last cards.

The reason for this is that it doesn't matter if the G3 is the last, the third-to-last, or the fourth-to-last card: Green can still be played out regardless of who draws G3 and G4 (or Green can't be played out if either is the last card). If the 3 is the second-to-last card, however, the player holding G5 needs to draw G3, and the other player needs to have G4.[/strike]

Nevermind, I neglected to take playing Y5 into account. I think the optimal play is to get florrat to draw a playable card? So florrat clues something, Notafool plays W5, then florrat clues Notafool if they drew something playable, and discards chop if they didn't (since that guarantees them drawing a playable card)?


2. Here's my optimal sequence (A="You", B="Partner"):
A: clue 5s
B: discard if M4 is A’s chop; play 5 if not
A: clue something if B discards; discard chop if B played 5
B: play 5 unless M4 is in A’s hand and isn’t chop; clue M4 if not A’s chop
A: clue something
B: play 5
A: play chop unless M4 was clued; play M4 if clued
B: play M5

Basically, B signals to A whether A's chop is to be played or discarded. Then, B plays as much as possible, cluing A about M4 at the appropriate time. Finally, when B draws the last card, A plays M4 into B's M5.


3. You/gamblingoof need(s) to draw the penultimate card in the deck; if M3 is the third-to-last or the last card in the deck, it doesn’t matter who draws it (similar to #1).


Nice puzzles, florrat! Do you have more?