A Winning Hanabi Strategy

Games

My favorite card game is Hanabi by Antoine Bauza. My friends and I have played it enough to learn a series of techniques that reliably boost our score. Some of these techniques were collected from the internet, but some are our own ideas. In this post, I’ll share what we’ve learned.

Notes to the Reader

I’ll assume that readers already understand the rules of the game. This post is not a tutorial, but a discussion of strategy.

Some of the fun of the game is discovering these techniques for yourself. If you haven’t attempted it yet, I recommend coming back to this post once you’ve spent some time exploring the game.

I suggest taking breaks as you read this post. Try introducing the techniques one at a time so you can internalize them without getting overwhelmed.

Example Players

I find that Hanabi is most fun with three players, so I’ll be using example scenarios involving three fictional participants: Alice, Bob and Charlie, sometimes abbreviated as A, B and C respectively. Assume the turn order is A → B → C → A etc.

Card Arrangement

One of the most important features of Hanabi is the constrained way in which players can communicate. The instructions explicitly state:

If you follow the rules closely, you can only communicate with your teammates when you give them information placing a blue token.

This rule clarifies that telling a teammate information without using a blue token is against the rules. The strictest interpretation implies that there can be no talking, asking questions, winking, or communication of any kind between teammates during the game. However, players should be free to strategize between games.

Let’s examine another rule that subtly changes our interpretation of the ‘blue token’ rule:

A player who is given information can rearrange [their] hand in order to put the cards concerned in an order which is easier for [them] to remember (on the left, on the right, further up or down).

From this sentence, we can conclude that the game was designed to accommodate the idea of card rearrangement. However, if all players are using the same techniques for arranging their hands, those techniques necessarily become an implicit means of communication. For example, if Alice sees Bob move a card within his hand, Alice now knows that Bob knows something new about his hand. This raises the question of whether this implicit communication breaks the ‘blue token’ rule. Because the above rearrangement rule is stated so specifically, I argue that the implicit communication by arrangement of cards is allowed.

Card Age

Now that we’ve decided card arrangement is allowed, let’s see what advantages we can get from it. I recommend arranging the cards by ‘age’, meaning that they’re sorted chronologically with your newest cards (i.e. most recently drawn) on your right and oldest cards (i.e. least recently drawn) on your left. The image below shows how your hand would appear to your teammates:

In this configuration, we can think of the newest and oldest card as each being in a special position. The newest card is called the ‘draw’ card. The oldest card is called the ‘chop’ card.

The idea here is to discard your cards in chronological order. The oldest card is called the chop because it’s next on the chopping block. This technique helps by making discards more predictable. If your teammate has a valuable card, you don’t need to worry about them discarding unless it’s on their chop. Conversely, you can safely leave your teammates with no time if they have a disposable card on their chop. The benefit from your own perspective is that you don’t worry about discarding your chop card, because your teammates would have told you if it was valuable.

The draw card is unique because the card in that position is always changing. We’ll see why that’s useful later.

Information Tracking

Sorting by card age is helpful for tracking cards that you know nothing about, but what about cards for which you’ve been given some information?

I recommend arranging these cards into another group in your hand. Maybe you leave a space between those cards and your ‘card age’ cards, or maybe you can hold on to the tops of the cards so they’re lower in your hand. There’s also no rule stating that you have to hold all cards with one hand, so for example, you could hold ‘card age’ cards in one hand, and the rest of your cards in another.

Once you’ve learned information about a card, you’ll want to have a strategy to distinguish that card from other cards with known information. I recommend sorting them by number if you know the number, and sorting by ROYGBIV if you know the color.

If you know both number and color, rotate that card 90° sideways.

In addition to rotating the cards sideways, observant players will notice that the design on the front and back of the cards is not symmetrical (at least it isn’t in the version of the game I bought). This means that you can track information about different groups of cards by turning some groups upside down. This idea applies nicely to the ‘card age’ section as well. For example, if Alice tells Bob he has one yellow card, Bob now knows about the yellow, but he also knows the rest of his hand is not yellow. He can flip all non-yellows upside down to track that information.

Efficiency

The remaining strategies all utilize the idea of information efficiency. In order to quantify efficiency in the context of Hanabi, let’s invent a unit of measure: units of information per blue token. Think of a single unit of information as being either the number or the color of a single card. For example, if Alice tells Bob that he has one white card, we can think of her turn as conveying one unit of information. For that one unit of information, Alice pays a price of one blue token. However, we can gain more efficiency by including a higher quantity of cards in a single turn. If Bob tells Charlie that he has two 1’s, he conveyed two units of information per token.

Gap of One

One of the most important tools for deduction in Hanabi is thinking about why a piece of information was given at the moment it was given. For example, if Alice tells Bob that he has a red card, Bob should think: “Why did Alice give that information at this time? The red card must be playable.”

If we take that idea one step further, we arrive at the ‘gap of one’ idea. The strategy works by giving information that cannot possibly be useful. Other players will ask themselves why that information was given at this time. The answer is that one of the players has a playable draw card that bridges the gap. Let’s use the following example scenario to explore some gap‑of‑one techniques:

If it’s Alice’s turn, she can use a technique called the finesse. First, Alice must notice two things: Bob has a playable draw card, and Charlie has the next card of the same color in his hand. To perform the finesse, Alice tells Charlie that he has one yellow card. Now it’s Bob’s turn, and he was paying attention to the information Alice gave to Charlie. Bob thinks: “Why did Alice give that information at this time? The information contained a gap of one, so the missing card must be on my draw.” Bob chooses to play his draw card. Now it’s Charlie’s turn, and he noticed that Bob played his draw card in response to Alice’s information. Charlie can infer that his card is a yellow 3. Charlie chooses to play his yellow 3 to complete the finesse.

Now let’s restart the above scenario, but it’s Bob’s turn instead. He can use a technique called the reverse finesse. First, Bob must notice two things: Alice has a playable draw card, and Charlie has the next card of the same color in his hand. To perform the reverse finesse, Bob tells Charlie that he has one 4 card. Normally, the giver of information wants the receiver of that information to play the relevant card. However, Charlie notices that the card is not playable, so he does something else on his turn. Alice notices the gap of one, and on her turn plays her draw card to complete the reverse finesse. Charlie notices Alice’s action, and infers that he has a green 4, which he can play on his next turn.

Let’s restart the scenario one more time, but now it’s Charlie’s turn. He can use a technique called the bluff. First, Charlie must notice two things: Alice has a playable draw card, and Bob has a gap‑of‑one card in his hand. To perform the bluff, Charlie tells Bob that he has one red card. Now it’s Alice’s turn, and she notices the gap of one and plays her draw card. To Alice, this situation was indistinguishable from a finesse until she played her card. Now it’s Bob’s turn and he notices that Charlie’s information caused Alice to play her draw card. Bob infers that Alice thought she had the red 2 on her draw, so his card must be a red 3.

In terms of efficiency, both finesse techniques are giving us 4 units of information per blue token. The bluff technique doesn’t actually convey the correct information to all players involved, but in functional terms, it achieves the same effect.

Inference

Let’s continue with techniques for increasing our information-giving efficiency using inference. Here’s another scenario:

It’s Alice’s turn, and she has rotated her yellow 4 sideways because she knows both the color and number on that card. Alice notices that Bob has a matching card on his draw, and decides to use a technique called the free discard. Alice discards the yellow 4 and gains a blue token. Now it’s Bob’s turn, and he notices that Alice knowingly discarded a playable card, so he plays his draw card. Bob inferred that Alice no longer needed her card because he had a matching card on his draw. In terms of efficiency, the free discard gains a blue token and conveys 2 units of information.

There’s another inference-based play that Alice could make on her turn: give information to Bob or Charlie about cards that can never be played. Bob and Charlie both have valuable cards on their chop. To save those cards, Alice tells Bob that he has three 1’s. Because a 1 card of each color is already on the table, Bob infers that he should move those cards to his chop and discard them in exchange for blue tokens. Similarly, Alice could tell Charlie that he has three green cards because the green stack is complete.

Let’s restart the scenario, but it’s Bob’s turn instead. Bob can initiate a play that combines what we’ve learned about the finesse and the free discard. First, Bob performs a finesse by telling Alice that she has one 3. Now it’s Charlie’s turn, and he infers that his draw card is a blue 2. Instead of playing his draw card, Charlie discards it. Alice notices that Charlie’s response to Bob’s information involved his draw card, so Alice’s card is probably a gap‑of‑one card. Alice plays her draw card because she infers that Charlie understood he had the blue 2, but discarded it because the other blue 2 is on her draw. It’s worth noting that coordinating these more complex plays involves a higher level of risk. For example, if Bob’s move was a bluff instead of a finesse, Charlie would have discarded a more valuable card. If executed successfully, this series of moves conveys 6 units of information at a net cost of 0 blue tokens.

Bob has another interesting option on his turn. He could initiate a color finesse. First Bob must notice that Alice has several cards of the same color, and that those color-matched cards are in order. Bob tells Alice that she has three blue cards. Alice rearranges her hand so the blue cards are together. Now it’s Charlie’s turn. Charlie notices that Bob’s information has rearranged Alice’s hand, so he tells Alice that she has two fours. Now it’s Alice’s turn and she ignores the yellow 4, because her card arrangement indicates she already knew that information. The other 4 card indicates a gap in her blue cards. Alice plays her blue 2, because she inferred that the gap cards were in order. This technique can range in efficiency between 2-5 units of information per blue token, depending on how many consecutive color-matched cards the player has in their hand.

More Exploring

I hope you’ve found this discussion of strategy helpful. There are undoubtedly more techniques that are possible in Hanabi, so I invite you to continue exploring the game.

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