Wikipedia:Reference desk/Archives/Mathematics/2015 October 28

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October 28

Shape name

What is the maths name for a "washer" - a circle with another smaller concentric circle removed? It is a sort of 2D torus. -- SGBailey (talk) 09:06, 28 October 2015 (UTC)

See Annulus (mathematics).

Note that the washer method is a variant of Disc integration used to calculate the volume of a hollow solid of revolution, where the integrand is the area of the annulus. -- ToE 09:15, 28 October 2015 (UTC)

What's a "punctured plane"?

The Annulus (mathematics) article says that the figure is equivalent to a punctured plane, and links to Glossary of topology. I thought I'd qualify the link to point to the correct entry, but there is none with that name. The closest appears to be "punctured neighbourhood". Is that the same concept? Rojomoke (talk) 12:55, 28 October 2015 (UTC)

I think "punctured" is a generic term meaning "with a point removed", used especially in homogeneous spaces where it doesn't matter which point is actually removed. So you can have a punctured line, a punctured plane, a punctured sphere, or a punctured torus, and all of these represent topologically different spaces. A punctured neighborhood would be similar but which topological type you get depends on the underlying space and on the neighborhood in question. As I see it the first problem is that the lead section in an article like that should be accessible to an average middle schooler, so talk about the topological homeomorphism class should be moved down toward the end. The second problem is that the meaning of "punctured" by itself should be added to the glossary, then you could give phrases like "punctured neighborhood" and "punctured interval" as examples. This is coming from my rather hazy memory of the terminology so it would be good to have a source in hand before making corrections. --RDBury (talk) 13:58, 28 October 2015 (UTC)

Agreed that "punctured plane" without much other context would usually be construed as $\mathbb {R} ^{2}\setminus \{p\}$, where p is often (without loss of generality) the origin. However Mathworld [1] seems the think that it should be the complex plane (which I think is too narrow a definition), and they do have a page on "punctured set" [2]. SemanticMantis (talk) 15:47, 28 October 2015 (UTC)

Yes, close enough. "Punctured" means "with a point removed", meaning specifically set subtraction. Best refs I can easily find online are on mathworld above, but just about any textbook ref would be preferable IMO. SemanticMantis (talk) 15:50, 28 October 2015 (UTC)

The statement "x is preferable to MathWorld" is usually true, without further knowledge of x. --Trovatore (talk) 17:36, 28 October 2015 (UTC)

Well ok, I mostly agree... But still, it sometimes has some useful bits of info that are not already on WP, and even though other sources are still preferable, they aren't always readily findable, accessible, and available :) SemanticMantis (talk) 19:44, 28 October 2015 (UTC)

Upper bound on the number of possible Scrabble games

I assumed this question had been canvassed before, but I couldn't find it in the Archives.

I know there's a fairly easily obtained exact answer to the question of how many possible chess games there can be. There's possibly a reasonably exact answer for the same question about Scrabble, but it seems the only way to be sure is to enumerate them all, and that would take far, far longer than the age of the universe, so it's not terribly what you might call practical.

However, surely we can at least come up with an upper bound.

Can we come up with an upper bound for the number of possible Scrabble games?

This says there are 5.716 × 10^300 possible valid Scrabble configurations.

But this gives the much lower figure of 69 × 10^180, and then excludes an unnamed proportion of them as being invalid, so it works out even lower.

These two results inhabit ball parks that are not even in the same galaxy, let alone the same city. Can anyone comment on them, and suggest the true value? Thanks. -- Jack of Oz ^{[pleasantries]} 21:26, 28 October 2015 (UTC)

Well, you can come up with a maximum number of moves by assuming just one tile is placed each turn and knowing the total number of tiles in the game (100). You could then start by assuming the tiles can be placed in any unoccupied square on the 15×15 board (although of course they can actually only be placed adjacent to certain tiles). That would provide a simple upper bound, although I'm sure we can bring it down quite a bit from there. You might get it down quite a bit further by considering that those 100 tiles are each only one of 26 letters (27 if you count the blanks separately) and that the first word must cover the center square. But, to really get a good estimate you would need to consider the number of valid Scrabble words and how many of those can be placed on the board in any given config. At this point you are doing a simulation. StuRat (talk) 21:36, 28 October 2015 (UTC)

Where did you get the idea there is a way to find the exact number of possible chess games? It's not true, we only have very rough estimates. -- Meni Rosenfeld (talk) 21:53, 28 October 2015 (UTC)

OK, thanks. Scrub that. -- Jack of Oz ^{[pleasantries]} 08:04, 29 October 2015 (UTC)

Actually, we do know the exact number of possible chess games ("possible" in the abstract sense; not necessarily physically realizable). It's ℵ₀. Of course for most of these games, one or both of the players necessarily had to pass up the opportunity to claim a draw based on repetition or the fifty-move rule — but you can do that. --Trovatore (talk) 17:37, 29 October 2015 (UTC)

Oh, and now I'm going to quibble my own quibble, because it occurs to me that I haven't dealt with the case of chess games that go on forever. Both players just keep on playing forever, and no one ever wins or claims a draw. Do we count those? In that case, the number of possible games is 2^ℵ₀. --Trovatore (talk) 17:42, 29 October 2015 (UTC)

This is no longer true. You're right that invoking the 50-move rule is optional, but another rule introduced in 2014, the 75-move rule, is automatic. The number of chess games was estimated by Hardy to be 10¹⁰⁵ (often misprinted as 10¹⁰⁵⁰), and the number of chess positions is estimated to be between 10⁴³ and 10⁴⁷. Jack, in your question it is important to distinguish between games and positions/configurations. Getting an upperbound on the number of configurations gives you no information on the number of games. Egnau (talk) 20:29, 29 October 2015 (UTC)

Rats. What spoilsports those FIDE people must be. Well, I can hardly be expected to keep up on rules changes that happened in the last two years. If it were up to me the rules would never change at all (though you would certainly be free to invent your own, very similar, game). --Trovatore (talk) 21:04, 29 October 2015 (UTC)