2048 with only 64 bits of state | Hacker News Reader

2048 with only 64 bits of state

github.com

173 pointstodsacerdoti43 comments

To me, the impressive part is implementing it in under 200 lines of bash script, not implementing it with only 64 bits of state. Nothing clever is required for 64 bits of state: a cell has 12 possible states (2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, and empty), which can be expressed in 4 bits, and 4*4*4 = 64.

The board rotation trick is very nice to achieve the short code length!

        w) squish ;;
        a) rot; squish; rot; rot; rot ;;
        s) rot; rot; squish; rot; rot ;;
        d) rot; rot; rot; squish; rot ;;

thank you i was so proud of it lmao

It's the first code snippet I've read that also works as modernist poetry.

>a cell has 12 possible states

There is 18 states. The final possible board state is 16 increasing power of 2s starting at 4, since it's possible for a tile to spawn in as 4. Then you also need states for 2 and empty making for a total of 18.

That's a special case. You could have a single bit for special case mode and then very few bits to say what happened.

Usually you can encode special states in a more compact form because the type of specialness is additional information meaning the single special bit is all you need to grow by.

A bloated form for the final state would be a bit to indicate specialness, 7 bits for specialness mode. End state mode is encoded as the turn before plus direction. So adding 10 bits in all, there's almost certainly enough in the knowledge that it is an end state to encode the board in 10 bits fewer, eliminating all expansion except for the specialness bit.

It's not a special case, it's a change of the rules to allow numbers greater than 2048. Which are allowed in some implementations, but not this one.

It's the default rules.

This is correct (if cells higher than 2048 are allowed, which varies by implementation.) But the game can still be represented in fewer than 16×18 bits, because not every board state is reachable. There can't be more than one cell with the maximum value. And if that is present, then there can't be more than one cell with the next-highest value, and so on. So you could devise some scheme to enumerate all reachable states and skip unreachable ones, and it would take fewer than 16×18 bits to indicate which one. The upper bound is at least bounded by ignoring representing the maximum value and then using 4 bits to indicate its necessarily singular position. There are also some other unreachable configurations, like if many copies of the same value are touching each other, since at least one pair of them would have been combined on the previous move.

You can have cells with more than one value with the value above 2048 but below 128K, and I don't think your scheme works much in those cases. It seems like an open question as to whether there are more than 2^64 possible states without the 2048 limit.

There are*

13 possible states, because the empty state is important, too. But they fit nicely into 4 bits.

It would be mildly interesting to super-package the state into the theoretically smallest possible amount of bits: 13 * 16 = 665416609183179841 possible states, which is approximately 2 * 59.2, so 60 bits would be enough, with some margin. A whole hex digit shorter!

That's what the code already does. It packs 16 fields into a base 12 representation that takes up 57.36 bits and uses the remaining 6.64 bits to store a random seed between 0 and 99 (exclusive).

I included empty in my count.

Is that really the theoretically smallest?

I think you can prove that some states representable by your proposal are unreachable. E.g. a board with 16 2s. So there might be another bit to spare.

It ought to be possible to generate valid states ordered by total value and lexicographical order or something.

But whether that is practical is a different matter. It does allow you to unambiguously define state #5546335 with the lowest number of bits

Although at that point, your attempts to cut the state will probably resemble a look-up table enumerating every valid board state, which obviously balloons the executable size by a large amount.

yes but there could conceivably be better tricks to discover

It's not quite literally the smallest, but the interesting question is what fraction of all states represent such unreachable states.

I was thinking "wow, how do you do that?" but then I forgot you're not dealing with any possible value between 1 and 2048 in a cell, only the powers of 2, so significantly fewer possible board states. Very cool.

I feel people often miss opportunities to map between (potentially complex, high entropy) state spaces into simple linear sequences of possible states and then either use those sequences to store the complex state data as a simple number or use the state of a system to encode data of some kind.

Like, if you use a limited 7-bit character set encoding for text and map that to a number in a sequence of possible orderings of a deck of 52 cards, you can store 32 characters (conveniently sized for passwords you might not want people to know are passwords).

Cool! I used a similar trick in my 2048 AI: https://github.com/nneonneo/2048-ai. In my case I allow the tiles to go all the way up to 32768, so I just have one tile per nybble.

Using this representation is great: it lets me pack a whole gameboard into a single machine register, which makes it super efficient. In this case I see that you’re able to pack in a seed value to enable replayability - neat!

Love the fact that you refer to your program as AI. We need more people to use AI to refer to non-LLM technologies lest the term be redefined.

Like others I found the concise implementation to be impressive! I have noticed a bug though. Using the "drive into the corner" strategy (I keep the high tile in the bottom left) sometimes the top left tile randomly gets a smaller value (e.g., goes from 16 -> 4) when I slide to the left.

Yeah, the first few times I thought maybe I was just imagining it; but I went from STATE=7280398215952, tried to merge my 64s into the left corner and instead ended up in STATE=7280398025476; my 64s have become 4s.

Can't reproduce it exactly as it happened, but if I run `STATE=7280398215952 bash 2048.bash` and press a 8 times, the 128 always becomes a 2.

thanks, i fixed it. it was only happening on bash 3.2 and i don't use a mac so i didn't see it. https://github.com/izabera/bitwise-challenge-2048/commit/73f...

Doesn't happen for me, i wonder if it's only on certain architectures or certain versions of bash or something

I saw it happen on bash 3.2.25.

It is worth mentioning that in the original game you can choose to keep going after you reach 2048. This game had to remove that option to achieve a 64bit state. Still, a clever implementation.

> keep going after you reach 2048. This game had to remove that option to achieve a 64bit state.

Since 15^16 < 2^64, you can still use a 64 bit state to reach 2^15=32768, which does not seem to be reachable in practice (the previous state would fill the whole board!)

I'm not sure how the math works out, but some googling suggests that 32768 is achievable in practice while 65536 and 131072 are theoretically possible but have only been achieved with undos.

edit: Explanation of the 131072 cap: https://puzzling.stackexchange.com/questions/48/what-is-the-...

Also, in one of the answers there someone claims to have achieved 65536 in practice

65536 has been achieved by 2 people without undos. However, it is very difficult as it’s only around a 7% chance with optimal play.

This is a partial video of one of them: https://youtube.com/watch?v=QQSLjPHg5P8

You need 16^16 for 32768 because 0 is used to represent an empty tile (2^0 = 1 is not used) which is exactly equal to 2^64.

The state in this implementation also stores a random seed (between 0 and 99, exclusive), so using 16^16 for the state would leave nothing for the random seed.

The LOC are really impressive and the implemention pretty smart. Thanks for sharing!

I'd love to see a minimal C program or similar, implemented as a single function `int64 run_2048(int64 state, int input)`, which takes the current state and an input (left, right, up, down), and returns the next state. Could be an interesting code golf exercise.

Total number of states = 11^16, assuming you don't reach 2048, just like you don't actually kill the king in chess.

Now log_2(11^16)=55.34, thus 56 bits are sufficient. That's tight!

Encode them in CJK and combining Unicode characters, to keep the visual representation as short as possible.

Wow,I have a bash script that just starts a docker container with some fancy arguments roughly the same LOC...

I too am impressed with his bash script. Simple and concise.

YOLO install command

  curl -fsSL https://raw.githubusercontent.com/izabera/bitwise-challenge-2048/develop/2048.bash -o "$HOME/bin/2048" && chmod +x "$HOME/bin/2048" && 2048

You could pin to a commit for safety

Thats a fair point

    curl -fsSL https://raw.githubusercontent.com/izabera/bitwise-challenge-2048/413abf35be0f47c4947a5ebd6d581355383041e1/2048.bash -o "$HOME/bin/2048" && chmod +x "$HOME/bin/2048" && 2048

Why not

    $HOME/.local/bin

Always fun to see what izabera has come up with - every single time I'm somewhere between delighted and terrified to see what she's made the computer do this time!

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