From: To: Subject: [eternity] Digest Number 9 Date: Friday, July 02, 1999 5:09 AM --------------------------- ONElist Sponsor ---------------------------- Campaign 2000 is here! http://www.onelist.com Discuss your thoughts; get informed at ONElist. See our homepage. ------------------------------------------------------------------------ There are 21 messages in this issue. Topics in today's digest: 1. R: [eternity] Sample Puzzle on website From: "Ing. Marco Polazzo" 2. Re: 201 pieces From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" 3. Re: Something I seem to recall From: "Ronald Stewart" 4. Repost: Hexagons From: "Mark Pursey" 5. Re: Against the grain From: "William Waite" 6. Re: So, who are you guys, anyway ? From: "Angel \"Java\" Lopez" 7. Re: [Re: [eternity] Opinion please] From: "Angel \"Java\" Lopez" 8. IMPORTANT WARNING for Eternity Screen Solver users From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" 9. Eternity Screen Solver v1.2 From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" 10. Re: Against the grain From: "Bob Harris" 11. Re: Eternity Screen Solver From: "Bob Harris" 12. Re: Eternity Screen Solver From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" 13. Re: Against The Grain From: dsaund2773@aol.com 14. Re: 201 pieces From: Andreas Gammel 15. Re: Against The Grain From: "Phillips, Roger" 16. Re: Eternity Screen Solver From: Andreas Gammel 17. Re: Eternity Screen Solver From: "Brendan Owen" 18. network of computers From: Andreas Gammel 19. Re: So, who are you guys, anyway ? From: lace@cix.co.uk (Mike Leigh) 20. Re: network of computers From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" 21. Re: Against The Grain From: "Mark Pursey" _______________________________________________________________________________ _______________________________________________________________________________ Message: 1 Date: Thu, 1 Jul 1999 14:48:44 +0200 From: "Ing. Marco Polazzo" Subject: R: [eternity] Sample Puzzle on website It took me 2 min. and 13 sec. Marco. -----Messaggio originale----- Da: Ronald Stewart [mailto:ron@ldstewart.freeserve.co.uk] Inviato: giovedì 1 luglio 1999 12.31 A: eternity@onelist.com Oggetto: [eternity] Sample Puzzle on website So, how long did it take everyone to complete the sample 6-piece puzzle at http://www.eternity-puzzle.co.uk/ ? It "only" took me 13 minutes and 28 seconds. I told you I was good! Good luck to all in this quest! - Ron _______________________________________________________________________________ _______________________________________________________________________________ Message: 2 Date: Thu, 1 Jul 1999 15:57:24 +0100 From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" Subject: Re: 201 pieces 201 pieces ???!!! Fine, this is the new Eternity Screen Solver Record ! May I ask your friend to send me a copy of the "BestPosition.ess" file located in the windows system directory of the machine that got the 201 pieces arrangement. I'll decipher it and will send him in exchange the complete layout of this grid filling. (I cannot offer more this time !... ;-) I'll probably will publish it on my new web site http://www.SwissKnifeSoftware.com (not yet ready at the moment, probably next week). Check for new release of E.S.S. there. P.F. Culand Mailto: pf_culand@bigfoot.com -----Message d'origine----- De : Andreas Gammel À : eternity@onelist.com Date : mercredi, 30. juin 1999 17:56 Objet : [eternity] 201 pieces >From: Andreas Gammel > >> 154 by Eternity Screen Solver v1.0 at >> http://lalsoft.hypermart.net/ScreenSolver.htm. > >a collegue here at the office just installed the screen saver and got a '201' >in only 10 minutes. >De we have a new record ? > >Andreas > > > > >--------------------------- ONElist Sponsor ---------------------------- > >Who is the most visited e-mail list community Web Service? >http://www.onelist.com >ONElist.com - where more than 20 million e-mails are exchanged each day! > >------------------------------------------------------------------------ > _______________________________________________________________________________ _______________________________________________________________________________ Message: 3 Date: Thu, 1 Jul 1999 15:29:21 +0100 From: "Ronald Stewart" Subject: Re: Something I seem to recall >From: Martin Watson > >When I first read about Eternity in the (London) Times back in the spring I >seem to recall Monckton saying that it could be possible to position 208 >pieces and find that the last one won't fit. This often happens with other >polyomino puzzles. I am only echoing this in case it is significant to >anyone writing software for a solution. > >Martin This was my problem with the six-piece puzzle. I think I found every possible variation of 5 pieces fitting! - Ron _______________________________________________________________________________ _______________________________________________________________________________ Message: 4 Date: Fri, 2 Jul 1999 01:23:56 +1000 From: "Mark Pursey" Subject: Repost: Hexagons *** sorry if this message has already been seen, but i think it got lost first time around *** Hi all, To test my program, i have been creating various hexagons from eternity pieces (to use as smaller puzzles), and have found that all sizes up to 64 pce can be built. I haven't yet found an 81 piece hexagon, but then it may take my solver a while to do so. Has anyone else been doing this, and if so, how high does this number go, the maximum possible being 196? (currently i am only looking with the grain). Also, has anyone encountered a largish shape which can be solved in more than one way with the same subset of [eternity] pieces, ie how is the case for multiple solutions looking? How about a regular shape that requires against the grain placement? In the rules for eternity it mentions a 'start' piece which is marked on the grid... any info on that? Mark Pursey / 27 / student / Sydney, Australia / yes i want the money _______________________________________________________________________________ _______________________________________________________________________________ Message: 5 Date: Thu, 1 Jul 1999 18:46:41 +0200 From: "William Waite" Subject: Re: Against the grain >Well, consider the trapezoid solution to the (grid-aligned) tridrafters (found by Miroslav Vicher) at: http://www.mathpuzzle.com/vichtd1.GIF >This solution has an "against-the-grain" component in it. As it turns >out (I've verified this with my program) -- ALL solutions to this trapezoid >have such a component. (There aren't many). > >Is Monckton's solution to Eternity all grid-aligned? I'm not sure anyone >knows for sure, but my guess is "no." I agree with Wei-Hwa. I think Monckton would try to be crafty and VERY PURPOSELY include a section against the grain. If we assume for a moment that he DID include a section against the grain, would that make it even more likely that most of the solutions (assuming there were more) would have against the grain sections? And a sideline question: even if there were millions of solutions (which seems possible) would one have better odds playing the lottery? William waite@get2net.dk _______________________________________________________________________________ _______________________________________________________________________________ Message: 6 Date: Thu, 1 Jul 1999 15:25:37 -0300 From: "Angel \"Java\" Lopez" Subject: Re: So, who are you guys, anyway ? >QUESTIONS >-------------- >What's your..... > >name : Angel "Java" Lopez >age : hmmm...... 37 >occupation : Software Developer >city : Buenos Aires >country : Argentina >email : ajlopez@ajlopez.com >homepage : under construction.... > >------- >Are you an active problem-solver, or just here out of general interest ? Generally, I'm interested in mathematical problems, and game strategics.... > >------- > > >------- >Do you hope that by listening to this list, you'll get the vital hint >that'll make you win the >prize ? No >------- > >------- >Do you own an Eternity puzzle ? No >------- > >------- >What's your best attempt by hand ? My intention is not use my hand, but my brain and my computer.... >------- > >------- >What's your best attempt by computer ? Under construction >------- > >------- >What language / system / pc are you using ? C/C++/Java >------- > >------- >What algorithms / heuristics do you use for Eternity >(brute force, backtracking, genetic algothims, simulated annealing, >SECRET) ? Genetic, backtracking, mixed >------- > >------- >Did you type in all 209 pieces yourself ? In this moment.... >------- > >------- >Did you try any simpler variations of Eternity ? Yes >------- > >------- >Do you know some other interesting web-sites ? No... www.mathpuzzle.com >------- > >------- >Other hobbies ? Life >------- > >------- >Other physical puzzles ? None >------- > >------- >Other programming projects ? Lisp interpreter.... Logo interpreter..... >------- > >------- >Are you willing to show the group your source code (after removing your >secret functions) ? No secret functions..... but the progress is slow...... >------- > >------- >Any other stuff you like to add ? Thanks for the list! >------- > >------- >Do you know any other interesting problems ? A lot..... not one in particular..... >------- > >------- >How are going to spend your 1000.000 pounds ? Beer, women, food... ;-) >------- > >------- >What's the meaning of life ? To make something that counts.... _______________________________________________________________________________ _______________________________________________________________________________ Message: 7 Date: Thu, 1 Jul 1999 15:29:55 -0300 From: "Angel \"Java\" Lopez" Subject: Re: [Re: [eternity] Opinion please] >From: david sprenkle > >I think a computer will crack it. I think it will be a creative algorithm >though, not just a straight brute force attempt. > Yes, a computer, or a NETWORK OF COMPUTERS..... a collaborative effort via Internet...... Angel "Java" Lopez ajlopez@ajlopez.com _______________________________________________________________________________ _______________________________________________________________________________ Message: 8 Date: Thu, 1 Jul 1999 22:07:25 +0200 From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" Subject: IMPORTANT WARNING for Eternity Screen Solver users IMPORTANT WARNING for Eternity Screen Solver users: I've just received my Eternity box and I double checked the layout of each pieces introduced in the E.S.S. v1.0 /v1.1... And unfortunately, I found 4 wrong pieces ! Damned !!!! I'm very sorry for the lost of your CPU time (and the winning chances) this error lead to... Of course a new version 1.2 of E.S.S. including the corrected (and triple-checked !) pieces will be soon available. (In one or two hours). To do as fast as possible, I'll send it directly to eternity@onelist.com as soon as it will be ready. I and Florian apologize again for the mishap... Please, if you distributed E.S.S. around you, transmit this warning too. P.F. Culand Swiss Knife Software P.S: The published Ed. Pegg/Vicher list was OK (except concerning the order of the pieces which is not exactly the same as the official Ertl's list: there are few inversions). The transcription errors were made by us during the edition of our piece list resource file ! Fortunately, no user sent us the "solution found" code at the moment !... :-) P.P.S: For future versions, stay tuned with our new distribution site nextly opened at http://www.SwissKnifeSoftware.com (from 5.7.99, I hope...). _______________________________________________________________________________ _______________________________________________________________________________ Message: 9 Date: Thu, 1 Jul 1999 23:09:52 +0200 From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" Subject: Eternity Screen Solver v1.2 [This message contained attachments] _______________________________________________________________________________ _______________________________________________________________________________ Message: 10 Date: Thu, 01 Jul 1999 18:35:22 -0500 From: "Bob Harris" Subject: Re: Against the grain I'm going to introduce some notation here. I'm going to start calling the length of the long side of a drafter N, the medium side S, and the short side T. These letters correspond to the angles Ninety, Sixty, and Thirty which are opposite those sides. N is one unit long, T is a half unit, and S is about .87 units. Note that people talking about parity have been uning L for Left parity, so L for long side would get confusing. from Wei-Hwa Huang: > Bob theorizes that any solution that has "against the grain" components will > have only 90-degree angles in those components. My reasoning is this: For a set of pieces to be placed against the grain, yet still be able to combine with other 'with-grain' pieces, the outline of the set must be one you can draw on the drafter grid. To be against the grain, the inside boundaries of the set (i.e. the boundaries among the set) must *not* align with the grid. To accomplish that, the outline must be able to be translated or rotated on the grid less than a full unit (a unit is the long side of a 30-60-90 drafter) and come to rest along grid lines. I couldn't see anway to do that except to have a rectilinear shape that can be shifted one half unit, say to the right. There are of course godzillions of such shapes, and some brief tests I did suggested that they can probably all be filled with eternity pieces as long as they are wide enough. Miroslav Vicher's images of misaligned subsections his solver stumbled into also supports this conjecture. I think he even had one such section enclosed within another. You (whoever is interested) can easily test this. Form such a region that's big enough to hold, say, eleven pieces. Don't make it too narrow. For example, take a 7N by 5S rectangle. Remove a 2N by 1S rectangle from the corner. This should have the desired area. Ask you solver to pack it. I'll bet you'll find a lot of solutions. Any of these solutions could potentially become an against-the-grain subregion in a solution of the puzzle. By the way, pieces 69 and 191 can be placed together to Clive Dawson: > Miroslav Vicher seems to believe these solutions > "cause trouble". Perhaps he would care to give us more details about > this. I think what he was referring to is that once his solver places a misaligned piece, it is forced to place a lot more misaligned pieces. If my conjecture is true (about the only misalignable shapes being rectilinear), then his solver has just forced itself into maintaining some rectilinear border within the puzzle. Chances are slim that he can close it out by random selction of pieces. And if he doesn't close it out, chances are that when he gets to the far edge of the puzzle, he'll have no solution. Unless he comes upon the edge of the puzzle which is in the direction matching his rectilinear region. Did any of that paragraph make sense? I've pretty much ignored misalignment in my search. Unless I find some parity-based argument that tells me that any solution will *have* to contain a misaligned region, I think the odds are that there are as many aligned solutions (relative to the size of the search space) as misaligned solutions. _______________________________________________________________________________ _______________________________________________________________________________ Message: 11 Date: Thu, 01 Jul 1999 18:41:35 -0500 From: "Bob Harris" Subject: Re: Eternity Screen Solver > 154 by Eternity Screen Solver v1.0 at > http://lalsoft.hypermart.net/ScreenSolver.htm. I downloaded this, but it appears I also need a 'screen saver component'. So I donwloaded that, but it appears I also need Delphi. What is Delphi? Is it free? Do I need it? Am I right, or can I just download the screen saver without having to load a bunch of other dominoes. _______________________________________________________________________________ _______________________________________________________________________________ Message: 12 Date: Fri, 2 Jul 1999 06:58:23 +0200 From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" Subject: Re: Eternity Screen Solver -----Message d'origine----- De : Bob Harris À : eternity@onelist.com Date : vendredi, 2. juillet 1999 01:13 Objet : Re: [eternity] Eternity Screen Solver >From: "Bob Harris" > > >> 154 by Eternity Screen Solver v1.0 at >> http://lalsoft.hypermart.net/ScreenSolver.htm. > >I downloaded this, but it appears I also need a 'screen saver component'. >So I donwloaded that, but it appears I also need Delphi. > >What is Delphi? Is it free? Do I need it? > >Am I right, or can I just download the screen saver without having to load a >bunch of other dominoes. You doesn't need anything else than the EternityScreenSaver.zip downloaded file. (please use the v1.2 version that I sent directly to the eternity@onelist.com distribution list rather than the old v1.0 available yet at http://lalsoft.hypermart.net/ScreenSolver.htm. - Unzip the two included files (EternityScreenSaver.scr and Readme.htm) - Read the instructions in the readme.htm file to install it. (its very easy) The Lalsoft Limited "screen Saver Component" is the Delphi component I used to tarnsform my Delphi program into a Windows screen saver. You doesn't need it to use the program. Delphi (from Borland/Inprise) is the developing tool I used to write my program. This is an Object Oriented visual developing tool based on Object Pascal. (An OO variation of the Pascal programming language). It is not free but is the best Windows compiler I know. Best regards, P.F. Culand. _______________________________________________________________________________ _______________________________________________________________________________ Message: 13 Date: 2 Jul 1999 05:42:53 -0000 From: dsaund2773@aol.com Subject: Re: Against The Grain I agree. If you wanted to make a unique puzzle you would want to introduce one or more sections that were against the grain. 1. What form would these clusters take? 2. Can we identify those pieces that could be used against the grain? 3. What if we worked on making one or more mini-puzzles from these pieces? I have also notice that there seemed to be groups of nearly identical pieces. These groups tend toward 6 or 12 pieces. Is the puzzle roughly comprised of six wedges? What about the underlying grid? Is it smaller by some factor? Someone mentioned this. Dick Saunders _______________________________________________________________________________ _______________________________________________________________________________ Message: 14 Date: Fri, 02 Jul 1999 08:05:22 +0200 From: Andreas Gammel Subject: Re: 201 pieces Pierre-François Culand wrote: > From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" > > 201 pieces ???!!! > > Fine, this is the new Eternity Screen Solver Record ! I must apologize, but this was a hoax! A friend of mine tricked me, because I always bugger on about this damn Eternity stuff. Before he could tell me I'd already sent it to the list. So the max is still 194 (I think ?) cheers Andreas _______________________________________________________________________________ _______________________________________________________________________________ Message: 15 Date: Fri, 2 Jul 1999 07:11:50 +0100 From: "Phillips, Roger" Subject: Re: Against The Grain Dick Saunders wrote: > I agree. If you wanted to make a unique puzzle you would want to > introduce one or more sections that were against the grain. In playing with the pieces by hand, I find I naturally go with the grain. In fact, I find it hard to try to introduce an out-of-phase section. I suspect that Chris Monckton developed the puzzle by hand, that he's relying purely on its complexity to make it difficult, and that he didn't "design" any other features into it. This is pure supposition, of course. -- Roger Phillips roger@spss.com "*wanigan* in a lumber camp, a chest for supplies or a kind of houseboat for loggers and their supplies" -- Chambers Dictionary on CD-ROM (1994) _______________________________________________________________________________ _______________________________________________________________________________ Message: 16 Date: Fri, 02 Jul 1999 08:49:57 +0200 From: Andreas Gammel Subject: Re: Eternity Screen Solver Thanks for the new version. I have a few suggestions for version 1.3: add options to - see the individual pieces actually apear on the screen (refresh less then 5 seconds). This would really look nice and make a lot more people download it - set the size of the board on the screen (too small now) - various color schemes to color the pieces (looks much nicer) - non-moving board and what would be really cool: - provide a choice of different algorithms - backtracking with settable number of look-aheads - use piece-segments that slide towards eachother - provide an algorithm interpreter that allows a logical algorithm to be typed in by the screensaver user (I have some ideas for this one, which I could send you) cheers Andreas _______________________________________________________________________________ _______________________________________________________________________________ Message: 17 Date: Fri, 2 Jul 1999 17:05:41 +1000 From: "Brendan Owen" Subject: Re: Eternity Screen Solver Hi All, I have another sudjestion display the full solution. :) Rgeards Brendan > > I have a few suggestions for version 1.3: > add options to > - see the individual pieces actually apear on the screen (refresh > less then 5 > seconds). This would really look nice and make > a lot more people download it > - set the size of the board on the screen (too small now) > - various color schemes to color the pieces (looks much nicer) > - non-moving board > > and what would be really cool: > - provide a choice of different algorithms > - backtracking with settable number of look-aheads > - use piece-segments that slide towards eachother > - provide an algorithm interpreter that allows a logical > algorithm to be typed > in by the screensaver user (I have some ideas for this one, which > I could send > you) > > cheers > Andreas > > > > > --------------------------- ONElist Sponsor ---------------------------- > > Looking to expand your world? > http://www.onelist.com > ONElist has 180,000 e-mail communities from which to choose! > > ------------------------------------------------------------------------ > > _______________________________________________________________________________ _______________________________________________________________________________ Message: 18 Date: Fri, 02 Jul 1999 09:10:09 +0200 From: Andreas Gammel Subject: network of computers Angel \"Java\" Lopez wrote: > From: "Angel \"Java\" Lopez" > >From: david sprenkle > >I think a computer will crack it. > Yes, a computer, or a NETWORK OF COMPUTERS. There are about 100.000.000 Pc's in the world now. For argument's sake, lets say they all are 1000 Mhz. Lets assume that a program could check an Eternity piece-setting each computer clock beat. This means one could check 100.000.000 x 1000 x 1000.000 = 100.000.000.000.000.000 = 10^17 possibilities per second. The big bang is about 15.6 billion years ago = 5*10^17 seconds. When all pc's ran since the beginning of time we can check in the order of 10^34 possibilties. So we only need 10^(209-34) = 10^165 entire universes to crack the problem.... Cool, let's do it, it's worth it ..... Andreas _______________________________________________________________________________ _______________________________________________________________________________ Message: 19 Date: Fri, 2 Jul 1999 08:57 +0100 (BST) From: lace@cix.co.uk (Mike Leigh) Subject: Re: So, who are you guys, anyway ? I would be very interested in your representation of the drafters Regards Mike Leigh _______________________________________________________________________________ _______________________________________________________________________________ Message: 20 Date: Fri, 2 Jul 1999 10:49:23 +0100 From: "=?iso-8859-1?Q?Pierre-Fran=E7ois_Culand?=" Subject: Re: network of computers This is true only if you consider the Eternity puzzle has only one solution. (Or if you want to find the Monckton's solution). But the Eternity puzzle has probably many (a lot of) different solutions. If this were not the case, it's clear that brute force would be useless, we would have much better chances by playing to the lottery. But, we can raisonably expect that the puzzle has many solutions. (In fact we would need something like 10^160 to 10^200 different solutions ! but there no reason not to believe so much different solutions exist.) Considering small puzzles, we observe that they generaly only have one or two solutions, but the more the size of the pieces is small regarding the total size of the grid, the more the puzzles have different solutions. So we can expect many solutions for the Eternity puzzle. Exploring the Eternity puzzle combination tree with brute force is like we were an ant climbing a (very big) cherry-tree seeking for a cherry. If there is only one cherry in the tree (or if we want to find only the "Monckton's cherry"), we will have to browse quite the entire tree to have a chance to reach it. But if the tree has a lot of different cherries regulary dispatched in the branches. We only need to browse a small bough to find our first cherry. And the first branches we choose to climb in the first nodes from the trunk don't matter (that means the first choosen piece placements) since the cherries regulary dispatched. What would be important to compute or to evaluate, is not the size of the complete cherry tree, but the "cherry density" it bears in its leaves. If the ratio of the total number of "cherries" divided by the number of leaves in the tree is about 10^34, you only need 1 universe duration to crack with your supposed cooperative 1000000 computers network. With Eternity Screen Solver exploring only 150 leaves/sec and guessing that 100000 computers would run it during one year, we would require a "solution density" of about 150*3600*24*365*100000 = 4.7*10^9 to get 50% chances to crack it. (Of course, this suppose that the solution density is regulary dispatched in the tree). How can we evaluate the "solutions density" ? Hum, a way to do it would be to first browse the complete tree and count the cherries and the leaves !!! Does somebody have a better solution ?... Waiting the answer let's run Eternity Screen Solver and cross your fingers... P.F. Culand. PS: Since the complete tree is very huge (and we know it is...) We don't need any cooperative strategy between the computers searching by brute force. the tree is big enough to be certain that two different randomly started sessions will never reach to the same explored bough. (The Eternity Screen Solver random seed is 32 bits long, this imply 4*10^9 different exploring areas in the tree, that's enough for the first year of search. Maybe in a future version will Iextend the seed to 64 bits...) -----Message d'origine----- De : Andreas Gammel À : eternity@onelist.com Date : vendredi, 2. juillet 1999 08:27 Objet : [eternity] network of computers >From: Andreas Gammel > >Angel \"Java\" Lopez wrote: > >> From: "Angel \"Java\" Lopez" >> >From: david sprenkle >> >I think a computer will crack it. > >> Yes, a computer, or a NETWORK OF COMPUTERS. > >There are about 100.000.000 Pc's in the world now. For argument's sake, lets >say they all are 1000 Mhz. >Lets assume that a program could check an Eternity piece-setting each computer >clock beat. >This means one could check 100.000.000 x 1000 x 1000.000 = >100.000.000.000.000.000 = 10^17 possibilities per second. The big bang is about >15.6 billion years ago = 5*10^17 seconds. When all pc's ran since the beginning >of time we can check in the order of 10^34 possibilties. > >So we only need 10^(209-34) = 10^165 entire universes to crack the problem.... >Cool, let's do it, it's worth it ..... > >Andreas > > > >--------------------------- ONElist Sponsor ---------------------------- > >How many communities do you think join ONElist each week? >http://www.onelist.com >More than 5,000! Create yours now! > >------------------------------------------------------------------------ > > _______________________________________________________________________________ _______________________________________________________________________________ Message: 21 Date: Fri, 2 Jul 1999 19:01:15 +1000 From: "Mark Pursey" Subject: Re: Against The Grain > From: "Phillips, Roger" > > In playing with the pieces by hand, I find I naturally go with the grain. > In fact, I find it hard to try to introduce an out-of-phase section. > I suspect that Chris Monckton developed the puzzle by hand, > that he's relying purely on its complexity to make it difficult, > and that he didn't "design" any other features into it. > This is pure supposition, of course. Bear in mind too that the human solver (haha) is expected to draw the pieces on a grid... if the solution is with the grain then each edge-line will be either gridpoint-gridpoint or gridpoint-midpoint. If the solution contains against the grain elements, then there will be edges that dont pass through any gridpoint, which would surely complicate things for people working by hand. If i was transcribing a solution onto paper, i would be cursing at this extra complication (not too much obviously) If against the grain solutions are ok, then what about solutions that involve a 30 degree rotation of pieces? There may (unlikely i admit, the area wouldn't add up) be solutions which involve this kind of rotation. 6 N edges is only 1% shorter than 7 S edges, and there's bound to be people (who haven't marked grids on their pieces) attempting wierd placements like this. Has anyone asked CM about grain directly? maybe he never intended for people to get stuck on this issue... BTW: are the 4 grain states in any way related to the 4 colour theorem? Mark _______________________________________________________________________________ _______________________________________________________________________________