In this amusing post, “Twelve Dimensional Chess is Stupid,” Steven Pav writes:
I cringe when I hear the term “Twelve Dimensional Chess” used as a metaphor. Certainly Twelve Dimensional Chess would be hard to visualize, and would present far more possible moves than regular two dimensional chess. However, high dimensional Chess suffers from a Curse of Dimensionality as the number of squares grows so quickly that play becomes uninteresting. In fact, I suspect that strategies exist which effectively guarantee a draw in sufficiently high dimensions. . . .
I [Pav] do not know the rules of high dimensional Chess, and I have assumed the players start with eight pieces and eight pawns. Maybe the number of pawns is linear, or even exponential in the dimension. Even so, it will take over 300 moves to promote the 64 pawns to Queens. Moreover, assuming one’s opponent could muster such an army without any losses, assembling such a large number of Queens in place to achieve checkmate might be tricky. Each Queen attacks at most 1.86 million squares. Again, this would be hard to visualize during play, but there are 2.2 billion internal squares (i.e. those not touching a boundary), and some 68.7 billion in total. Which means that even if your opponent has dozens of Queens on the board, each Queen can attack only a small fraction of the available squares. You could move your King largely at random without coming under attack.
So “Twelve Dimensional Chess” as a metaphor for a situation requiring great foresight or strategy in the face of many possible decision is flawed. Instead, it is a more apt metaphor for a very lonely random walk punctuated by infrequent interactions with others you can easily dodge.
In high-dimensional space, no one can hear you scream.
I honestly can’t recall the last time I heard anyone refer to high-dimensional chess, usually eleven-ty dimensional, as anything but a way to make fun of someone who thought they were being highly strategic when it was clear to everyone else that they just fooling themselves.
Nkh:
The quoted post is from several years ago, so maybe the use of the term has changed. For some reason, I always thought of “12-dimensional,” not “11-dimensional,” chess. Back when I was a kid they’d talk about 3-dimensional chess, but, y’know, inflation.
I kind of recall people talking about 3-d chess as a serious thing at some point. I don’t actually pay much attention to chess, but I got the impression that it doesn’t work very well as a game. So I had always assumed any higher dimension was at least poking fun at the idea. And that’s just how I have always heard it used since. And the dimension gets sillier as time goes on.
Andrew: “3-d chess” was popularized by the original Star Trek series, since there would be scenes of it being played. It’s not just normal chess where everything moves in another axis.. The rules were never specified, but the board is definitely different to make it not merely normal chess plus more dimensionality.
https://chessantiques.com/product/star-trek-tridimensional-chess-set-franklin-mint/
“a way to make fun of someone who thought they were being highly strategic when it was clear to everyone else that they were just fooling themselves.”
That is a perfect summation of how the term is used now, at least in my experience. I do recall the term being applied unironically to Elon Musk a few years ago, back when it still seemed possible that he really was an “autodidact polymath” (his words). The newer usage fits him far better.
Never heard the 11 or 12 version, but familiar with 4- and 5D version. Weirdly, heard it meant to mean both nkh’s version, *and* it’s positive opposite, someone who’s thinking seems silly/inscrutable but who is actually several steps ahead of everyone. [So both defs easily apply to Musk, in different ways :) ]
But the biggest take-home I took from that, is that it’s author seems to want to be the second coming of Neil DeGrasse Tyson. In his pendantage, his essay became almost self-referential.
> pendantage
That word’s gotta be at least 12 dimensional.
Ha! Well played 🙂👍🏽
https://en.wikipedia.org/wiki/5D_Chess_with_Multiverse_Time_Travel?wprov=sfti1
Seems like real life where an intractable combinatorics problem needs to be addressed with heuristics like incorporating past experience and observing frequencies. I’m not sure that makes it uninteresting.
First, that depends on how you define the objective. If the objective is something like: maximize the chance you can put the other king into check (say if they do make a mistake) it will still be a hard problem.
Moreover, because you have many squares where you can move your king without being threatened doesn’t actually tell us much. There are all sorts of famous mathematical problems where you can delay losing almost arbitrarily long and seem to have a huge degree of freedom but still are guaranteed to lose: see for instance the hydra problem.
Also this analysis makes some controversial assumptions about how you generalize the rules. Here is an equally good generalization:
Rook; select one component of your 12d coordinate which remains fixed while you set the other 11 to any value.
Bishop: Add n* v where v is a vector whose elements (in Cartesian basis) are all of the form +-1 to your position and n is an integer.
Queen:. Combo of rook/bishop
I could continue but I think the point is clear. There are many ways to generalize chess to more dimensions which avoid this issue.
Peter:
Yeah, good point. There are lots of ways to generalize. That said, the way Pav describes it is kind of how I was picturing the (hypothetical) game.
Andrew: People have come up with playable rules for the Star Trek 3-d version, see e.g. discussion here
https://www.chess.com/forum/view/chess-variants/star-trek-tri-dimensional-chess-ever-play-it