Two questions about Snake
Obviously, I haven't been posting much here lately. Here's another entry for the "problem dump" series -- two math problems I don't really intend to work on (at least not at the moment). I'll probably put this up on my website later.
Actually, I'm a little surprised to find that I haven't written anything about this here earlier. Oh well. Here we go.
Let's consider the game of Snake; except instead of playing it on a grid, we're going to play it on an arbitrary finite [simple] graph. We assume the snake starts at length 1 -- the game begins by the "computer" (who acts adversarially) picking a starting spot for the snake and a starting spot for the fruit. When the snake consumes the fruit (and thereby grows by 1), the computer picks a new spot for the fruit, which cannot be somewhere currently covered by the snake. We'll say the player wins if the snake covers the whole graph. If the game goes on infinitely, I guess that's a draw, but we'll give it to the computer.
(Here are some obvious ones -- if a spanning subgraph of a graph is winnable, then so is the whole thing; any cycle is winnable, so so is any graph with a Hamilton cycle; in order to be winnable, a graph must have a Hamilton path. But like I said, I'm not going to put all my notes here.)
So the general question then is, for which graphs can the player always win? (Hence why I said draws go to the computer.) Now this question is not necessarily that concrete; there might not be any nice characterization. I have bunch of notes about necessary conditions and sufficient conditions for this, but I'm not going to list them all here. I do, however, have two specific questions about this that I'd like to ask.
So -- you'll notice I didn't give a totally formal specification of the rules above, and there's a reason for that. When you formalize the rules, you realize there's the question: Should a snake of length 2 be allowed to turn around? As in, move its head to the current location of the tail. If you write the rules in the obvious manner, this is entirely legal, but it seems kind of unintuitive. So we can have two variants of the game: The weak game, where this is allowed; and the strong game, where it is not. (Note that you could consider graphs with multiple edges, and say that then in the strong game you're allowed to turn around if there's a double edge; but I'm restricting to simple graphs for reasons you'll see shortly.)
There is at least one graph that is weakly winnable but not strongly winnable, namely, a path of length 3. So, question number one: Are there any other such graphs? I suspect the answer is no but have been unable to prove it. Hence the restriction to simple graphs -- if I'm right, then unless the "underlying simple graph" is P3, adding multiple edges simply won't affect anything. (And it's easy to see how it affects P3.)
Here's a second question. When I initially mentioned this problem to John, he suggested that whether a graph is winnable or not should depend only on its topology. But this is not so; I have examples of (weakly or strongly) winnable graphs that can be subdivided to be made non-winnable. However, I have been unable to find an example of a (weakly or strongly) non-winnable graph which can be subdivided to be made winnable. So, question number two: If a graph is non-winnable (weakly or strongly), is the same true of any subdivision of it? For this question I don't really have a good idea what the answer should be. My guess would be yes, but I wouldn't be all that surprised if someone found a counterexample.
By the way, someone -- I forget who, unfortunately -- suggested that you could also play Snake on infinite graphs, with the win condition being that you grow arbitrarily large (now it's only a "draw" if the game goes on infinitely but your size stays bounded). But that seems like a pretty different problem, so I'm not going to consider that here.
-Harry
Actually, I'm a little surprised to find that I haven't written anything about this here earlier. Oh well. Here we go.
Let's consider the game of Snake; except instead of playing it on a grid, we're going to play it on an arbitrary finite [simple] graph. We assume the snake starts at length 1 -- the game begins by the "computer" (who acts adversarially) picking a starting spot for the snake and a starting spot for the fruit. When the snake consumes the fruit (and thereby grows by 1), the computer picks a new spot for the fruit, which cannot be somewhere currently covered by the snake. We'll say the player wins if the snake covers the whole graph. If the game goes on infinitely, I guess that's a draw, but we'll give it to the computer.
(Here are some obvious ones -- if a spanning subgraph of a graph is winnable, then so is the whole thing; any cycle is winnable, so so is any graph with a Hamilton cycle; in order to be winnable, a graph must have a Hamilton path. But like I said, I'm not going to put all my notes here.)
So the general question then is, for which graphs can the player always win? (Hence why I said draws go to the computer.) Now this question is not necessarily that concrete; there might not be any nice characterization. I have bunch of notes about necessary conditions and sufficient conditions for this, but I'm not going to list them all here. I do, however, have two specific questions about this that I'd like to ask.
So -- you'll notice I didn't give a totally formal specification of the rules above, and there's a reason for that. When you formalize the rules, you realize there's the question: Should a snake of length 2 be allowed to turn around? As in, move its head to the current location of the tail. If you write the rules in the obvious manner, this is entirely legal, but it seems kind of unintuitive. So we can have two variants of the game: The weak game, where this is allowed; and the strong game, where it is not. (Note that you could consider graphs with multiple edges, and say that then in the strong game you're allowed to turn around if there's a double edge; but I'm restricting to simple graphs for reasons you'll see shortly.)
There is at least one graph that is weakly winnable but not strongly winnable, namely, a path of length 3. So, question number one: Are there any other such graphs? I suspect the answer is no but have been unable to prove it. Hence the restriction to simple graphs -- if I'm right, then unless the "underlying simple graph" is P3, adding multiple edges simply won't affect anything. (And it's easy to see how it affects P3.)
Here's a second question. When I initially mentioned this problem to John, he suggested that whether a graph is winnable or not should depend only on its topology. But this is not so; I have examples of (weakly or strongly) winnable graphs that can be subdivided to be made non-winnable. However, I have been unable to find an example of a (weakly or strongly) non-winnable graph which can be subdivided to be made winnable. So, question number two: If a graph is non-winnable (weakly or strongly), is the same true of any subdivision of it? For this question I don't really have a good idea what the answer should be. My guess would be yes, but I wouldn't be all that surprised if someone found a counterexample.
By the way, someone -- I forget who, unfortunately -- suggested that you could also play Snake on infinite graphs, with the win condition being that you grow arbitrarily large (now it's only a "draw" if the game goes on infinitely but your size stays bounded). But that seems like a pretty different problem, so I'm not going to consider that here.
-Harry