|
Main
Date: 27 Nov 2007 06:55:49
From: Berkeley Brett
Subject: Disproof: Possible Number of Chess Games
|
Hello all: How many possible Chess games are there? Well, I think I've come up with an easy, intuitive disproof of a number one often sees. Perhaps some of the mathematically savvy readers of this group can give their opinion of my claim. (Thanks in advance, should you choose to do so.) Here's a number I often see: "In a game of 40 moves, the number of possible board positions P(40) is at least 10^(120) according to Peterson (1996)...." http://mathworld.wolfram.com/Chess.html (other estimates are given in the article hyperlinked above) 10^(120) is a 1 followed by 120 zeros For comparison, one estimate of the number of particles in the observable universe is 10^(80) -- a much, much smaller number. ( http://varatek.com/scott/bnum.html ) Well, the 10^(120) estimate is, I think, demonstrably too large (by far). Here's my reason for thinking so: In a game of Chess, a square can be in only one of 13 states. It is either 1) empty, 2) occupied by one of the 6 white pieces (King, Queen, Rook, Bishop, Knight, Pawn), or 3) occupied by one of the 6 black pieces (King, Queen, Rook, Bishop, Knight, Pawn). Since there are 64 squares, this means there are 13^(64) imaginable states of the Chessboard. Of course, 13^(64) is itself an overestimate, since there are not enough pieces to "populate" all the squares with all possible combinations (and even if there were, not all possible combinations are legal). But what we can tell from this consideration is that any number that is GREATER than 13^(64) is obviously too high! Now, just how big is 13^(64)? Well, my Microsoft Windows calculator gives the estimate 1.96053...e+71 (a 72-digit number) Assuming that is a reasonable estimate, this number is far, far smaller than 10^(120)!! Q.E.D. If my disproof is sound, may I dub the number 13^(64) "Caissa's Constant," being a fixed overestimate of the number of possible Chess positions. By the way, Mathworld, is an excellent resource for all subjects mathematical (and all subjects Mathematica!) -- it is well worth a bookk, in my view. http://mathworld.wolfram.com/ Your feedback is most welcome.... -- David Brett Richardson http://www.100bestwebsites.org/ "The 100 finest sites on the Web, all in one place!" Widely-watched non-profit ranking of top Internet sites
|
|
|
Date: 28 Nov 2007 12:33:02
From: Stephan Bird
Subject: Re: Disproof: Possible Number of Chess Games
|
On Wed, 28 Nov 2007 11:30:35 -0600 in [email protected], Kenneth Sloan <[email protected] > wrote: > so, it's more important for the answer to be interesting than for it to > be correct? Well, as we've shown, the actual number of games is infinite, so it's probably more interesting for the answer to be 'interesting' rather than correct. (Or something like that) Stephan -- Stephan Bird MChem(Hons) AMRSC Currently in Caernarfon, Wales
|
|
Date: 28 Nov 2007 07:50:55
From: [email protected]
Subject: Re: Disproof: Possible Number of Chess Games
|
Although this is a degenerate example, it is correct: the number of possible chess *games* is infinite, as is easily demonstrated by one of many possible examples. Consider the sequence 1. Nf3 Nf6 2. Ng1 Ng8 Now repeat this sequence any arbitrary number of times. Though the number of positions generated is just a few, the number of *games* possible is as many as you wish, even though the games are not exactly thrillers. A draw can be claimed by repetition or the 50 move rule under this scenario, but that is an option, not a requirement, as I understand the rules (otherwise my example is incorrect).
|
| |
Date: 29 Nov 2007 10:24:27
From: Ray Johnstone
Subject: Re: Disproof: Possible Number of Chess Games
|
On Wed, 28 Nov 2007 07:50:55 -0800 (PST), "[email protected]" <[email protected] > wrote: >Although this is a degenerate example, it is correct: the number of >possible chess *games* is infinite, as is easily demonstrated by one >of many possible examples. Consider the sequence > >1. Nf3 Nf6 >2. Ng1 Ng8 > >Now repeat this sequence any arbitrary number of times. Though the >number of positions generated is just a few, the number of *games* >possible is as many as you wish, even though the games are not exactly >thrillers. > >A draw can be claimed by repetition or the 50 move rule under this >scenario, but that is an option, not a requirement, as I understand >the rules (otherwise my example is incorrect). That is why I wrote: A "game" must here be defined as a legal sequence of moves ending in mate, stalemate or a draw by obligatory imposition of the three-fold repetition and fifty-move laws so that the result is beyond dispute. [email protected] www.iinet.com.au/~ray
|
| | |
Date: 29 Nov 2007 11:42:14
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
|
Ray Johnstone <[email protected] > wrote: > A "game" must here be defined as a legal sequence of moves ending in > mate, stalemate or a draw by obligatory imposition of the three-fold > repetition and fifty-move laws so that the result is beyond dispute. Giving the players the option of not claiming possible draws does not put the result into dispute. Under the actual laws of chess, these are all legal games, where `**' means any one of `Black resigns 1-0', `White resigns 0-1' and `Draw agreed 1/2-1/2': ** 1.Nf3 ** 1.Nf3 Nf6 ** 1.Nf3 Nf6 2.Ng1 ** 1.Nf3 Nf6 2.Ng1 Ng8 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 {Black announces his intention to play 6... Ng8. Draw by repetition} 1/2-1/2 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 Ng8 ** 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 Ng8 {White claims a draw by repetition} 1/2-/12 1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 Ng8 {White announces his intention to play 7.Nf3. Draw by repetition} 1/2-1/2 and so on. It's just that there are infinitely many of them, so the question is trivial and boring without the supposition that draws will be claimed when available. Dave. -- David Richerby Mentholated Dish (TM): it's like www.chiark.greenend.org.uk/~davidr/ a fine ceramic dish but it's invigorating!
|
| |
Date: 28 Nov 2007 16:29:19
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
|
[email protected] <[email protected] > wrote: > Although this is a degenerate example, it is correct: the number of > possible chess *games* is infinite [...] A draw can be claimed by > repetition or the 50 move rule under this scenario, but that is an > option, not a requirement, as I understand the rules (otherwise my > example is incorrect). You are absolutely correct that the number of possible chess games is infinite because draws by repetition or the fifty-move rule have to be claimed and are not automatic. In order to generate a non-trivial problem, one supposes that either player will claim a draw as soon as he has the opportunity. One then asks how many of these `interesting' games there are. Dave. -- David Richerby Hilarious Projector (TM): it's like a www.chiark.greenend.org.uk/~davidr/ 16mm film projector but it's a bundle of laughs!
|
| | |
Date: 28 Nov 2007 11:30:35
From: Kenneth Sloan
Subject: Re: Disproof: Possible Number of Chess Games
|
David Richerby wrote: > [email protected] <[email protected]> wrote: >> Although this is a degenerate example, it is correct: the number of >> possible chess *games* is infinite [...] A draw can be claimed by >> repetition or the 50 move rule under this scenario, but that is an >> option, not a requirement, as I understand the rules (otherwise my >> example is incorrect). > > You are absolutely correct that the number of possible chess games is > infinite because draws by repetition or the fifty-move rule have to be > claimed and are not automatic. > > In order to generate a non-trivial problem, one supposes that either > player will claim a draw as soon as he has the opportunity. One then > asks how many of these `interesting' games there are. > > > Dave. > so, it's more important for the answer to be interesting than for it to be correct? -- Kenneth Sloan [email protected] Computer and Information Sciences +1-205-932-2213 University of Alabama at Birmingham FAX +1-205-934-5473 Birmingham, AL 35294-1170 http://KennethRSloan.com/
|
| | | |
Date: 01 Dec 2007 18:59:39
From: Guy Macon
Subject: Re: Guy Macon on the longest possible chess game
|
Kenneth Sloan wrote: > >David Richerby wrote: > >> [email protected] <[email protected]> wrote: >> >>> Although this is a degenerate example, it is correct: the number of >>> possible chess *games* is infinite [...] A draw can be claimed by >>> repetition or the 50 move rule under this scenario, but that is an >>> option, not a requirement, as I understand the rules (otherwise my >>> example is incorrect). >> >> You are absolutely correct that the number of possible chess games is >> infinite because draws by repetition or the fifty-move rule have to be >> claimed and are not automatic. >> >> In order to generate a non-trivial problem, one supposes that either >> player will claim a draw as soon as he has the opportunity. One then >> asks how many of these `interesting' games there are. > >so, it's more important for the answer to be interesting than for it to >be correct? David Richerby correctly anwered to question asked, noted that the question was not interesting, and then posed a related and far more interesting question. I fail to see on what basis you have a problem with that. BTW, do you have an answer (or at least an upper limit) to David Richerby's question? -- Guy Macon <http://www.guymacon.com/ >
|
| | | | |
Date: 02 Dec 2007 02:04:48
From: Phil Carmody
Subject: Re: Guy Macon on the longest possible chess game
|
Guy Macon <http://www.guymacon.com/ > writes: [SNIP - whatever] If you wish to be taken seriously on this usenet lark, then don't munge subject lines so. Narcissism is neither big nor clever. It's tempting to re-title the thread "his experience sucking donkey balls", just to see what you'd do with it. Phil -- Dear aunt, let's set so double the killer delete select all. -- Microsoft voice recognition live demonstration
|
| | | | | |
Date: 02 Dec 2007 11:44:17
From: Guy Macon
Subject: Re: Guy Macon on the longest possible chess game
|
Phil Carmody wrote: >If you wish to be taken seriously on this usenet lark, >then don't munge subject lines so. Narcissism is neither >big nor clever. Thanks for bringing this up. I welcome criticism and am very much willing to change my behavior to suit your preferences where appropriate. May I explain why I acted as I did? It may be that you will see what I am getting at and agree -- or at least be willing to tolerate two unnecessarily words in a subject line. As for my general reason for changing the subject line, the original was incorrect and labeled as such by the original poster in the thread. I believe that it was proper to change it from the number of possible games (which is not what he posted about) to what he did post about (the number of possible positions) or to what another poster brought up in reply (the longest possible game being infinitely long and thus the number of possible games being infinite). If your objection is to any such change of subject lines, I am open to hearing your arguments and will do as you wish if the arguments are compelling -- but I suspect that the real objection is to the first two words in the changed subject line, not changed subject lines in general. As for my reason for putting my name in the subject line, it is in response to a situation that you most likely are not aware of. A while back, an individual in one of the Linux newsgroups came to the incorrect conclusion that one of his many anonymous enemies was, in fact, me. As far as I can tell, his only reason for thinking that is the fact that I was posting through Supernews -- the largest news service as far as generating text-based messages goes. I switched to posting from Giganews, but that didn't help. This individual then decided to work toward his stated goal of making me unhirable by salting Google with material that would drive away potential employers. His chosen methods included posting copies of my resume modified to make me look bad, posts from nonexistent employers claiming I was fired for arson, accusations of breaking laws that don't exist, and impersonating me. These posts to Usenet ended up on the many websites that repost content from newsgroups. (Your name has ended up on such websites as well; see http://www.google.com/search?Phil+Carmody -- but without any obvious forgeries.) While I am employed at the moment, I may be looking for work some time in the future, and I have a handicapped wife who needs a lot of expensive medical care. Because of this I decided to attempt to drive the forgeries lower in the Google search results. Alas, the Google search algorithm puts posts with a person's name in the title much higher than posts by that person, and thus the only practical way to accomplish that goal is to put the name in the subject line. I would prefer not to do so, because it may make others see what you saw -- narcissism -- but given the alternative I would prefer to have someone think me a narcissist rather than think me a person fired for arson. >It's tempting to [snip] just to see what you'd do with it. I would very much prefer that you not do so. I certainly wouldn't do such a thing to you, after seeing the damage that can be done to a person's online reputation firsthand. -- Guy Macon <http://www.guymacon.com/ >
|
| | | | | | |
Date: 03 Dec 2007 11:13:16
From: David Richerby
Subject: Re: Guy Macon on the longest possible chess game
|
Guy Macon <http://www.guymacon.com/ > wrote: > I believe that it was proper to change it from the number of > possible games (which is not what he posted about) to what he did > post about (the number of possible positions) or to what another > poster brought up in reply (the longest possible game being > infinitely long and thus the number of possible games being > infinite). But the thread isn't about the longest possible game: it's about the number of possible games, on the assumption that draws will be claimed ASAP. I realise that knowing the longest possible game under that assumption is useful but it's hardly the whole answer to the question. So your subject line is no better than the original. > As for my reason for putting my name in the subject line, it is in > response to a situation that you most likely are not aware of. > [viz: large numbers of forged postings] I'm sorry you're being attacked. But thank you so much for making your problem our problem as well. How about, instead of engaging in pissing contests with an idiot, you just inform potential employers that there's a lot of forged material in your name on the net? Your response to the situation means that there are two major sources of information about you on the net. One, which is forged, makes you look bad. The other, which is not forged, makes you look like an immature narcissist. I wouldn't want to have to explain the latter to a potential employer. Dave. -- David Richerby Portable Microsoft Gerbil (TM): www.chiark.greenend.org.uk/~davidr/ it's like a children's pet that's really hard to use but you can take it anywhere!
|
| | | | | | |
Date: 02 Dec 2007 11:10:09
From: Kenneth Sloan
Subject: Re: Kenneth Sloan on Guy Macon on the longest possible subject line
|
Sounds like a great justification. Can I play, too? Guy Macon wrote: > Phil Carmody wrote: > >> If you wish to be taken seriously on this usenet lark, >> then don't munge subject lines so. Narcissism is neither >> big nor clever. > > Thanks for bringing this up. I welcome criticism and am > very much willing to change my behavior to suit your > preferences where appropriate. May I explain why I acted > as I did? It may be that you will see what I am getting > at and agree -- or at least be willing to tolerate two > unnecessarily words in a subject line. > > As for my general reason for changing the subject line, the > original was incorrect and labeled as such by the original > poster in the thread. I believe that it was proper to > change it from the number of possible games (which is not > what he posted about) to what he did post about (the number > of possible positions) or to what another poster brought > up in reply (the longest possible game being infinitely long > and thus the number of possible games being infinite). If > your objection is to any such change of subject lines, I am > open to hearing your arguments and will do as you wish if > the arguments are compelling -- but I suspect that the real > objection is to the first two words in the changed subject > line, not changed subject lines in general. > > As for my reason for putting my name in the subject line, > it is in response to a situation that you most likely are > not aware of. A while back, an individual in one of the > Linux newsgroups came to the incorrect conclusion that one > of his many anonymous enemies was, in fact, me. As far as > I can tell, his only reason for thinking that is the fact > that I was posting through Supernews -- the largest news > service as far as generating text-based messages goes. > I switched to posting from Giganews, but that didn't help. > > This individual then decided to work toward his stated goal > of making me unhirable by salting Google with material that > would drive away potential employers. His chosen methods > included posting copies of my resume modified to make me > look bad, posts from nonexistent employers claiming I was > fired for arson, accusations of breaking laws that don't > exist, and impersonating me. These posts to Usenet ended > up on the many websites that repost content from newsgroups. > (Your name has ended up on such websites as well; see > http://www.google.com/search?Phil+Carmody -- but without > any obvious forgeries.) > > While I am employed at the moment, I may be looking for > work some time in the future, and I have a handicapped > wife who needs a lot of expensive medical care. Because > of this I decided to attempt to drive the forgeries lower > in the Google search results. Alas, the Google search > algorithm puts posts with a person's name in the title > much higher than posts by that person, and thus the only > practical way to accomplish that goal is to put the name > in the subject line. I would prefer not to do so, because > it may make others see what you saw -- narcissism -- but > given the alternative I would prefer to have someone think > me a narcissist rather than think me a person fired for > arson. > >> It's tempting to [snip] just to see what you'd do with it. > > I would very much prefer that you not do so. I certainly > wouldn't do such a thing to you, after seeing the damage > that can be done to a person's online reputation firsthand. > Kenne -- Kenneth Sloan [email protected] Computer and Information Sciences +1-205-932-2213 University of Alabama at Birmingham FAX +1-205-934-5473 Birmingham, AL 35294-1170 http://KennethRSloan.com/
|
| | | | |
Date: 01 Dec 2007 14:57:09
From: Kenneth Sloan
Subject: Re: Guy Macon on the longest possible chess game
|
Guy Macon wrote: > Kenneth Sloan wrote: >> David Richerby wrote: >> >>> [email protected] <[email protected]> wrote: >>> >>>> Although this is a degenerate example, it is correct: the number of >>>> possible chess *games* is infinite [...] A draw can be claimed by >>>> repetition or the 50 move rule under this scenario, but that is an >>>> option, not a requirement, as I understand the rules (otherwise my >>>> example is incorrect). >>> You are absolutely correct that the number of possible chess games is >>> infinite because draws by repetition or the fifty-move rule have to be >>> claimed and are not automatic. >>> >>> In order to generate a non-trivial problem, one supposes that either >>> player will claim a draw as soon as he has the opportunity. One then >>> asks how many of these `interesting' games there are. >> so, it's more important for the answer to be interesting than for it to >> be correct? > > David Richerby correctly anwered to question asked, noted that the > question was not interesting, and then posed a related and far more > interesting question. I fail to see on what basis you have a problem > with that. > > BTW, do you have an answer (or at least an upper limit) to David > Richerby's question? > > Your question is correct - but not interesting. -- Kenneth Sloan [email protected] Computer and Information Sciences +1-205-932-2213 University of Alabama at Birmingham FAX +1-205-934-5473 Birmingham, AL 35294-1170 http://KennethRSloan.com/
|
| | | |
Date: 29 Nov 2007 11:29:53
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
|
Kenneth Sloan <[email protected] > wrote: > David Richerby wrote: >> In order to generate a non-trivial problem, one supposes that >> either player will claim a draw as soon as he has the opportunity. >> One then asks how many of these `interesting' games there are. > > so, it's more important for the answer to be interesting than for it > to be correct? It's not incorrect as long as you state the problem you're solving correctly: `the number of possible chess games, assuming that both players will claim a draw as soon as they have that option, is N.' Dave. -- David Richerby Adult Peanut (TM): it's like a roasted www.chiark.greenend.org.uk/~davidr/ nut that you won't want the children to see!
|
|
Date: 28 Nov 2007 11:26:05
From: Ray Johnstone
Subject: Re: Disproof: Possible Number of Chess Games
|
On Tue, 27 Nov 2007 06:55:49 -0800 (PST), Berkeley Brett <[email protected] > wrote: I've written a note about the number of chess games at: http://members.iinet.net.au/~ray/Chessgames.htm [email protected] www.iinet.com.au/~ray
|
|
Date: 27 Nov 2007 16:54:01
From: Anders Thulin
Subject: Re: Disproof: Possible Number of Chess Games
|
Berkeley Brett wrote: > Assuming that is a reasonable estimate, this number is far, far > smaller than 10^(120)!! So ... you're not computing the same number. Did you check out the reference to Peterson 1966 to see what he was referring to? The number of different chess positions is generally stated to be on the order of 2*10^43 (no promoted pieces), or the order of 10^50 (promoted pieces), or 10^?? (including promoted pieces, castling state, e.p. state and 50-move clock state.) They are, of course, approximations. The number that Peterson mentioned was the number of different positions in all games of a length of 40 moves. That is not clearly one of the numbers above. (The number is just mentioned, without any source reference or indication of how it was computed.) -- Anders Thulin anders*thulin.name http://www.anders.thulin.name/
|
|
Date: 27 Nov 2007 08:41:42
From: Berkeley Brett
Subject: Re: Disproof: Possible Number of Chess Games
|
Thanks to both David and Mike for their feedback. My confusion occurred to me just after posting the message -- just as I often see a Chess blunder milliseconds after making it! Oh well! Sometimes our mistakes can also be instructive and valuable! -- David Brett Richardson http://www.100bestwebsites.org/ "The 100 finest sites on the Web, all in one place!" Widely-watched non-profit ranking of top Internet sites
|
|
Date: 27 Nov 2007 15:19:48
From: Mike Robson
Subject: Re: Disproof: Possible Number of Chess Games
|
Le 27-11-2007, Berkeley Brett <[email protected] > a �crit�: > Hello all: > > How many possible Chess games are there? > > Well, I think I've come up with an easy, intuitive disproof of a > number one often sees. Perhaps some of the mathematically savvy > readers of this group can give their opinion of my claim. (Thanks in > advance, should you choose to do so.) > > Here's a number I often see: > > "In a game of 40 moves, the number of possible board positions P(40) > is at least 10^(120) according to Peterson (1996)...." > > http://mathworld.wolfram.com/Chess.html > > (other estimates are given in the article hyperlinked above) > > 10^(120) is a 1 followed by 120 zeros > [...] > Your feedback is most welcome.... > > -- > David Brett Richardson > http://www.100bestwebsites.org/ > "The 100 finest sites on the Web, all in one place!" > Widely-watched non-profit ranking of top Internet sites Quite right. People often sloppily confuse the number of possible games (for which 10^120 looks like a conservative estimate) with the number of possible board positions (which is much smaller because the same position can be reached in a vast number of different ways.) I have seen "between 10^30 and 10^40" quoted as an approximation to the number of possible board positions. When I tried to calculate a better estimate I decided it was near to the bottom of that range (~10^39) but I no longer have more than a hazy memory of how I came to that conclusion. Mike.
|
|
Date: 27 Nov 2007 15:56:14
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
|
Berkeley Brett <[email protected] > wrote: > Well, I think I've come up with an easy, intuitive disproof of a > number one often sees. Perhaps some of the mathematically savvy > readers of this group can give their opinion of my claim. (Thanks > in advance, should you choose to do so.) Your argument that 13^64 is an upper bound on the number of ways to arrange the pieces on the board is correct. However, this in no way proves that the total number of games cannot be more than 13^64. Many positions can be reached in more than one way. For example, the position after 1.e4 e5 can be reached by all kinds of routes, including but not limited to, 1.e4 e5 1.e3 e6 2.e4 e5 1.Nf3 Nc6 2.Ng1 Nb8 3.e4 e5 Thus, the number of possible games dramatically exceeds the number of possible positions. Dave. -- David Richerby Accelerated Permanent Hat (TM): it's www.chiark.greenend.org.uk/~davidr/ like a hat but it'll be there for ever and it's twice as fast!
|
|
Date: 27 Nov 2007 07:18:46
From: Berkeley Brett
Subject: Re: Disproof: Possible Number of Chess Games
|
Ah, of course, this is the number of possible Chess POSITIONS --not Chess GAMES! Nevermind! -- Brett http://www.100bestwebsites.org/ "The 100 finest sites on the Web, all in one place!" Widely-watched non-profit ranking of top Internet sites
|
| |
Date: 27 Nov 2007 16:35:24
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
|
Berkeley Brett <[email protected] > wrote: > Ah, of course, this is the number of possible Chess POSITIONS --not > Chess GAMES! Gaaaaah, this whole thread is a pile of confusion about the difference between positions and games. :-) I've filled in Mathworld's comment form to point out that their page is incorrect. Dave. -- David Richerby Disgusting Laptop Goldfish (TM): it's www.chiark.greenend.org.uk/~davidr/ like a fish that you can put on your lap but it'll turn your stomach!
|
|