A new 2048-style challenge

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Yes, the game 2048 is one of those tablet/cellphone games which also run on a desktop browser which have fascinated numerically-inclined people such as myself. It is a wonderful waste of time, something to do while you are doing something else and actually need to pull yourself away for a few moments and clear your mind by diverting your attention to something else for a bit.

As many of you know, it is normally played on a 4×4 grid. When you search for “2048 game” on the Duck Duck Go search engine, at the top of the selection is a game Duck Duck Go has provided:

Duck Duck Go’s attempt to replicate the “classic” game of 2048

My cellphone app offers me a variety of grid dimensions to play on: 4×4, 5×5, 6×6, and 8×8. The bigger the grid size, the easier it gets to obtain the total 2048. Once you obtain that tile, you win, although you can continue playing. You can play until you have no more room to move or merge your tieles. If you run out of room before you reach 2048, you lose.

The 8×8 tileset is so easy, you can have a mess on the grid and still reach 2048 eventually, with plenty of room left on the board to move your tiles around. So if you are just starting out, you might find it easier to play with the larger boards, eventually graduating to the 4×4 board once you have a clear idea what is going on, and have built a sense of intuition. My version of the game allows you to undo your last turn (but no more than that) in case your last move was less than ideal. It is also useful because sometimes I could swipe right and the tiles move in the wrong direction for some reason. They are supposed to move in the direction of the swipe.

Out of curiosity, I went one size down and tried out the 6×6 mode of the app, and so far, I have two 4096 tiles, and a 16384 tile with no end in sight:

A bit blurry, but there are always a ton of empty squares in this game, and the game board, as you can see is less than ideal for tile distribution. But I’m still getting 16384 and beyond. This same game has been played and stored off and on for the past week.

So, since 2048 is fairly easy on grids beyond 6×6, I began to wonder what numbers could pose a challenge on these larger squares. For 8×8, I think I found what might be a suggestion:

Uhhh, woah.

By the title of this game, they appear to be suggesting that I try to reach a tile labelled “2,147,483,648”, a 10-digit number just over 2 billion. I would be curious to see how they achieved fitting such a number on a tile once it is obtained. But how long would that take? On a 4×4 grid, getting that tile could take a couple of hours, if you get there at all.

The original 2048 suggests that 2048 be the winning tile, which is a power of 2, namely 2^{10}. Taking the base-2 log of 2,147,483,648 on my calculator, I find that this new number is equal to 2^{31}. I am beginning to think that I will need to pass this on to my children in my will, assuming I haven’t lost the game yet.

2147483648 is a game offered currently on GitHub, so it is likely the side project of a professional programmer or a pet project for a hobbyist and could still contain bugs (imagine the nightmare in testing and debugging this game into the high numbers). It appears that the only one developing the project is an anonymous, nameless, faceless programmer going by the handle “jiangtyd”. That said, it seems to come with clever innovations, such as allowing the program to auto-move the tiles on its own to create interesting combinations. You can also change base to something like 3, or increase the tiles by way of the Fibonacci sequence. The “3” selection is a bit lame, in that the successive tiles that need to be formed are really 3\times 2^n, so you are really dealing with powers of 2 again, except in a different way.

What are your thoughts?