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Posts: 23951
Joined: September 20th, 2002, 8:30 pm

2048 game

QuoteOriginally posted by: outrunThere is a nice thread at Stackoverflow on strategies, a lot of heuristics and ideas but nothing mathematical yet. Shouldn't we be able to do something here?The math would seem to be illusive due to the complexity if the combinatoric and constraints.The 4x4 topology, DOF of the tile slides, and rules for pair-merger seem to define a pretty messy space.That said, the total number of 1024-512-256-128-64-32-16-8-4-2 snakes isn't that large after the various reflections and rotations are removed. There's only 3 possible locations for the 1024 tile (the other 13 are reflections or rotations) and then an average of 1 to 3 possible locations for the next smaller adjacent tile. And a lot of those possible adjacencies lead to invalid snakes. My rough guess is there's only a few hundred unique snakes.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

2048 game

2x2: 4 initial states, 2 first moves, 3 subsequent next-tiles, 3 or 4 next moves, 2 or 3 subsequent next-tiles,...then it gets messy because the sequence branches depending on whether cells got mergedGood point about the N->1. The permutations of all the gaps and merger opportunities in the four rows makes N quite high.The space of accessible game boards seems quite large, though.(Side brainteaser: what is the LOWEST score possible in this game? There's an easy theoretical answer based on the lowest-score "game-over" configuration of checkerboard of 4 & 2 but is that configuration reachable????)

rmax
Posts: 6080
Joined: December 8th, 2005, 9:31 am

2048 game

It collapses to a simple state, where if by chance the game generated 2,4,2,4,2,4 in that order and placed them on the board where you would rotate so that they would move but not collate. It is almost an entropy like problem as that configuration would have high entropy.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

2048 game

Methinks reaching the 2,4,2,4,... checkerboard might be quite hard. It's really hard to move tiles around and really easy to create positions that have one or more unavoidable (and undesirable) pair-merger.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

2048 game

QuoteOriginally posted by: outrunwhat's the minimal board dimensions that will allow you to play indefinitely? (my contribution to the next set of Millenium problems)Hmmm... I'm pretty sure you need at least N cells to create a game that gets to 2^N.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

2048 game

QuoteOriginally posted by: outrunThe 4 operators that map board-states to new states (N->M mapping with M<=N) remind me of 4-arry version of a Collatz treeExactly! And one of the issues that makes 2048 hard is that one does not always have a choice on the operator. There are many board states in which one or more of the operators because unusable for game play and one is forced to pick one of the remaining operators which may disrupt the pattern one has worked so hard to create.P.S. Collatz (aka Hailstone sequences) was a worse time sink that 2048 game for me.

tagoma
Posts: 18378
Joined: February 21st, 2010, 12:58 pm

2048 game

I reached 11644 points.

tagoma
Posts: 18378
Joined: February 21st, 2010, 12:58 pm

2048 game

QuoteOriginally posted by: MHillDamn! Screwed up at 57332My "snake" got split by a rogue tile, and then again as I was working around it.No more!impressive. Paul will ring you in a minute and organize an antidoping test.

MHill
Posts: 489
Joined: February 26th, 2010, 11:32 pm

2048 game

QuoteOriginally posted by: tagomaQuoteOriginally posted by: MHillDamn! Screwed up at 57332My "snake" got split by a rogue tile, and then again as I was working around it.No more!impressive. Paul will ring you in a minute and organize an antidoping test.I've been going cold turkey since then. I could spend a lifetime chasing another high like that, but inside, I know I'll never get another high quite like it. It's not out my system yet though - I keep feeling compelled to visit this thread. I should probably go hang out in a different community to keep me away from temptation.

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...

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