As you get into higher dimensions, you get exponentially more space, which makes getting to 2048 easier and easier (3x3x3 3D suffers from this, somewhat). So 5D would probably need to bump the winning score up to 4096 or even higher, but then the game start taking far too long.

this 4d is actually harder since each dimension is 2x2. 7052 :/ still haven't beaten it..

Yes, but the smallest interesting 5d one is 2x2x2x2x2, i.e. 32 squares, which is much more flexible than the 16 squares of the original (and of this one).

So 2048 in 2048 dimensions after that?

Unfortunately, even assuming a board diameter of 2, storing that game's any state would require some 10^435 times more bits than there are cube Planck lengths in the observable universe.

Only naively. Since the user is incapable of inputing that number of bits, it does not require that number of bits to store. In this case, one need not even be that fancy; a fairly naive sparse algorithm would do fine. It would take a user a long time to have enough tiles laid down for the time to calculate the new board state to even be perceptible, and the computer will still be able to compute the next board state faster than a human can comprehend the effects of the previous move.

Your far, far, FAR bigger problem is UI. :) Again, you need not display all dimensions at once, so you can treat it as an n-dimensional problem, but that still calls for some crazy UI pretty quickly.

Undoubtfully the number is really large, but I'm not sure what exactly you mean by cube Planck lengths. A comparison to atoms would be more clear.

Still, it wouldn't have to. Since most arrays would have pieces with lower numbers( 2,4,8 ) you could very efficiently compress the data.

At some point you would run out of memory but it would be playable. Just like those Game of Life implementations with seemingly infinite grid.

Planck length is a theoretical very small unit of length in physics. It is the scale of quantum foam and several orders of magnitude smaller than all those fancy particles like quarks, electrons, protons and atoms: https://en.wikipedia.org/wiki/Orders_of_magnitude_%28length%...

BTW you are right about compressing: you could store the initial state very efficiently, and probably even play the to the end without trouble.

if you store it with log2(n) you could easily store everything with a single byte. That brings the storage requirements down considerably. Though you still have huge issues with storage, not to mention input. For input you'd need 4096 combinations!

You can even do 2 numbers per byte (11 takes 4 bits).

Sorry for the downvote, I mis-clicked. Wish you could change a vote afterward.

Or 2048 in one dimension.

2048 in l^2 Hilbert space.

Still countable. [/troll]

Seems easier than 4D: I played on a 16-square single board as if it were 4D, and the "pulled down" numbers from the other (essentially random) board if they lined up nicely, or if I needed new numbers.