Hunting the Invisible 1

[Travis Hall]

There’s an interesting tactical problem that I’ve run into in Hanabi games a few times recently.

It is Alice’s turn. Bob’s turn will follow, then Charlie’s turn, and then Dan’s turn. There is no fifth player. (Hence, four-card hands.)

The current score is 27. Bob holds two coloured 5s, both marked. Charlie holds the 1 black and three non-black cards, and none of Charlie’s cards are marked. There is exactly one token available, and no misfires have occurred. There is exactly one card remaining in the draw deck. No relevant Flamboyants are available. (Either Flamboyants are not in use, or the two remaining Flamboyants are those that give tokens.)

Charlie is an extremely intelligent and experienced player, and can be assumed to figure out very complex chains of logic, but the players have not set a convention to deal with this specific scenario. All have demonstrated ability to both give and follow finesses and bluffs, including advanced variations such as reverses, long finesses (and long partially-reversed finesses), trash bluffs, and even hidden finesses. They have not established any of the conventions that are more unusual on BGA.

How does Alice clue in order to give the best possible chance of achieving a score of 30 at the end of the game?

Note that if Alice plays or discards, Bob will get only one more turn, so cannot play both coloured 5s. Alice must clue, or the probability of achieving 30 drops to 0. So, positional discard hints are impossible. Likewise, Bob and Charlie must both play. Nobody cares what Dan does.

Now, the answer is trivial if Charlie holds only a single 1. But what if Charlie holds x 1r 1k x? x 1r x 1k? In these scenarios, cluing 1s would give at best a 50/50 chance of Charlie picking the correct card to play.

I have my own answer to this problem, and I have used it successfully at least once, but at least one other time “Charlie” expressed confusion and did not play the correct card. I will let others ponder and perhaps attempt to solve this before I explain my approach. (Maybe someone will have a better idea than I did.)

[Raal]

As this is situation near end game, we can use clues in unstandard ways.
Charlie can know who own 1k by easy guessing after Bob's turn, so I will use number clue just meaning position as below

x 1r 1k x -> clue 3
x 1r x 1k -> clue 4

[Travis Hall]

As this is situation near end game, we can use clues in unstandard ways.
Charlie can know who own 1k by easy guessing after Bob's turn, so I will use number clue just meaning position as below

x 1r 1k x -> clue 3
x 1r x 1k -> clue 4

Interesting thought! I hadn’t considered cluing to indicate the position.

I see some weaknesses, though. What if Charlie’s hand is 3r 1r 1k x? Might Charlie read meaning into the mark to his leftmost card? I guess you can use Dan’s hand as target, because like I said, nobody cares about Dan’s cards at this point. But do we then run into problems with 3s in particular positions in Dan’s hand? What if there are such difficulties in Bob’s hand as well?

Also, is it possible Charlie might count from the right, instead of from the left?

Are there ways to improve this approach? Are there better alternatives?

[Travis Hall]

(The observation that Charlie can know, without any other indication, that he is holding 1k is, of course, absolutely vital to solving this problem.)

[Travis Hall]

Okay, so, time for my solution to the problem.

My approach has been to find ways to clue different scenarios that will make sense, noting that when the scenario is harder to clue, the easier scenarios will be eliminated by the fact that I don’t use the simpler clue. So, in order...

If Charlie holds precisely one 1: Clue 1. The correct card to play will be marked, and Charlie plays it.

If Charlie holds two 1s...

... 1k 1 x x, or 1k x 1 x: By familiar conventions, we can clue 1s, and Charlie will play the leftmost. (We could even clue nothing, and expect Charlie to play left.)

... x 1k 1 x: By familiar conventions, we can clue 1s, and Charlie will play the leftmost.

... 1 1k x x: Now things are getting tricky. We can’t clue nothing and allow Charlie to just play leftmost. We can’t clue 1s either. What can we do instead?

Clue colour to the leftmost card. Because we haven’t clued 1s, Charlie knows it isn’t any of the prior scenarios. Because I’ve marked the leftmost card with a colour, Charlie knows there is something important about the fact that that card is coloured. If it’s not a 1, we don’t need to tell him about it - we only care about 1s. So it must be a 1, and it must be the wrong 1. So where is the 1k? Charlie’s best guess is the next card along. He plays second from left, correctly.

... x 1 1k x: We still can’t clue 1s, or let Charlie play blind. Clue colour to the left 1. Similar logic applies, so Charlie plays the next card along from the colour.

(Ah, but what if the leftmost card matches the extra 1’s colour? Say, r 1r 1k x, or m 1 1k x. Just mark both, and Charlie will assume the next card along is correct.)

... 1 x 1k x: Still can’t let Charlie play blind, or clue 1s. If we clue the extra 1 by colour, by the previous logic, Charlie will play the next card along, which is wrong. We have to do something else. So, clue the second card from the left by colour. Charlie can’t tell the difference between this and x 1 1k x, so treats it the same, and plays the next along from the colour. That is correct.

... but what if the card second from left is black? 1 k 1k x. Then we can’t clue it by colour. So, clue by number instead. Charlie knows we would rather clue 1s directly if possible, but we don’t, so we can’t. He knows we would clue the extra 1 by colour if we could get the right play that way, but we don’t, so we can’t. So the fact that we clue by a number other than 1 tells Charlie his left card is a coloured 1, and his second card is a black non-one. He plays the next along, which is 1k.

I have deliberately avoided discussing 1k on chop scenarios so far. We have to face that possibility, though. So...

... x x 1 1k: Following previous logic, clue colour to the extra 1.
... x 1 x 1k: If the third card from left can be clued by colour, do so. Previous logic applies. Charlie doesn’t have to know that the colour card is not actually a 1.
... x 1 k 1k: Clue third card by number. Previous logic applies.
... 1 x x 1k: Still clue third card by colour, or number if not possible.

Finally, 1k x x 1: Noting that our system has to have other ways to clue a 1k on the right (possibly lying about whether the indicated card is a 1), cluing 1s to show 1 x x 1 would be reserved for this scenario. In any other configuration that allows this clue, there’s a less-ambiguous alternative. So this would draw 1k from the left. Or, Alice can waste the clue and not touch Charlie’s cards, and Charlie should guess left anyway.

Especially problematic scenario:
2m 1g 2r 1k: Here, if we clue red, there’s a hole in the resulting pattern. Same if we clue 2s. Which unmarked card is 1k? But if 1k was in second position, we could just clue 1s, so 1k should be on the right. There’s a few similar configurations, but I haven’t spotted one where the same logic does not apply.

And then we go on to three 1s and four 1s scenarios, but they can be covered by just cluing 1s when the leftmost 1 is black, or cluing colour to the left of 1k otherwise, or other number to the left of 1k if there is a black card to the left of 1k.

And that covers all possible scenarios. I can’t find a case where there isn’t a clue to fit. Can you?

[Raal]

I see some weaknesses, though. What if Charlie’s hand is 3r 1r 1k x? Might Charlie read meaning into the mark to his leftmost card? I guess you can use Dan’s hand as target, because like I said, nobody cares about Dan’s cards at this point. But do we then run into problems with 3s in particular positions in Dan’s hand? What if there are such difficulties in Bob’s hand as well?

I see, this approach should clue no card.
Maybe, clue for Bob is also good choice.
There is still weakness of possibility to play leftmost card.


I notice the approach of using Dan's hand.

Charlie's: x x x 1k
Dan's: x x x 3r

Clue card just bottom of 1k; clue red or 3.
It's available in rare condition.

Also, is it possible Charlie might count from the right, instead of from the left?

In the cases "1k x x x" and "x 1k x x", There are better approachs to tell it.
Therefore, Charlie would consider cases of "x x 1k x" and " x x x 1k".
It feels nonsense that clue 1 to tell rightmost card, because it can be applied only when 1k is lonely rightmost 1.

[master of bluff]

hi its simple just use trash push or trash pull to indicate where is 1k
trash pulls and push are written conventions that we are using xd

[Romain672]

So what is trash push and trash pull in short:
- if you want slot 1, you use things like trash bluff (clue green on g1 while g4 is on stacks and g5 isn't played, to make next player play his slot 1, which will tell the card is trash)
- if you want slot 2/3(/4), you clue the leftmost card of it, as a known trash clue (like if r5, m5, and k2 are played, on a hand b1-r3-k1-b4, you clue red to make k1 played)(named trash pull)
- if you want the second chop card, you can clue the chop, as a known trash clue (named trash push)

I don't think I ever see them in bga though.

[Travis Hall]

hi its simple just use trash push or trash pull to indicate where is 1k
trash pulls and push are written conventions that we are using xd

You need to read my description of the original problem again. “They have not established any of the conventions that are more unusual on BGA.” So it doesn’t help that you are using trash push and trash pull conventions, because Alice, Bob, Charlie, and Dan have not established them. And neither have many other players on BGA.

That said, and considering Romain’s description of these techniques, the solution I presented incorporates trash pull, along with the logic that leads to that convention. So if you’re a smart enough group of players, you can get that same effect without relying on a convention established in advance.

Also, even in the case when the 1k is leftmost in Charlie’s hand, trash bluff isn’t really doable when all remaining playable cards except the 1k are visible to all. Any attempt just looks like marking trash.