From: To: Subject: [eternity] Digest Number 15 Date: Wednesday, July 07, 1999 3:21 AM --------------------------- ONElist Sponsor ---------------------------- Attention ONElist list owners. http://www.onelist.com We've just added a "NO ATTACHMENTS" option. See homepage for details. ------------------------------------------------------------------------ There are 25 messages in this issue. Topics in today's digest: 1. Re: So, who are you guys, anyway ? From: Russell McCormick 2. Re: Parity From: "Ronald Stewart" 3. Eternity hints From: David Clark 4. Re: So, who are you guys, anyway ? From: =?iso-8859-1?q?Richard=20Heath?= 5. Piece 34 starter position From: David Clark 6. Re: who are you? From: "Richard Burchill" 7. Re: Piece 34 starter position From: "Mark Pursey" 8. Re: Piece 34 starter position From: cphipps@synetics.co.uk 9. Start a programming thread. From: "Richard Burchill" 10. Re: Piece 34 starter position From: "Ronald Stewart" 11. Nice try... From: falk.werner@zurichre.com 12. Another one... From: falk.werner@zurichre.com 13. And a third one... From: falk.werner@zurichre.com 14. By the way: Re: Who are You ? From: falk.werner@zurichre.com 15. a quick (maybe) question From: MasterMB@aol.com 16. Re: Marc Becker and orientation issue From: Dsaund2773@aol.com 17. Re: a quick (maybe) question From: "Mark Pursey" 18. quick (maybe) question and an 'off-grid' exercise From: path@multipro.com.au 19. Re: quick (maybe) question and an 'off-grid' exercise From: path@multipro.com.au 20. status + complexity From: Andreas Gammel 21. the ambiguous hexadude From: "Mark Pursey" 22. Re: status + complexity From: cphipps@synetics.co.uk 23. Re: status + complexity From: Andreas Gammel 24. Re: Welcome to eternity@onelist.com From: Chris McLaren 25. Re: status + complexity From: "Ronald Stewart" _______________________________________________________________________________ _______________________________________________________________________________ Message: 1 Date: Tue, 06 Jul 1999 19:46:31 +0900 From: Russell McCormick Subject: Re: So, who are you guys, anyway ? At 13:31 28/06/99 +0200, you wrote: >>>> Hi all, we just welcomed our 76th mailing list member. And still going strong. I thought it would be a nice idea to get to know the group a bit better. That's why I composed a little list of questions. I'd like you to fill it out, and send it to the list. just answer any question you feel like, but if you don't like it, then don't, its a free world... Andreas QUESTIONS -------------- What's your..... name : Russell McCormick age : 27 occupation : programmer city : Adelaide country : Australia email : russell.mccormick@iname.com homepage : ------- Are you an active problem-solver, or just here out of general interest ? ------- Active (when I get the time) ------- Do you hope that by listening to this list, you'll get the vital hint that'll make you win the prize ? ------- One can always hope...... ------- Do you own an Eternity puzzle ? ------- Nope....I have one on order though ------- What's your best attempt by hand ? ------- 0 ------- What's your best attempt by computer ? ------- 154 (Screen Saver) ------- What language / system / pc are you using ? ------- haven't decided yet.....still working on structure ------- What algorithms / heuristics do you use for Eternity (brute force, backtracking, genetic algothims, simulated annealing, SECRET) ? ------- may the (brute) force be with me (with some optimisations) ------- Did you type in all 209 pieces yourself ? ------- no ------- Did you try any simpler variations of Eternity ? ------- only on eternity web site and mathpuzzle.com ------- Do you know some other interesting web-sites ? ------- sure ------- Other hobbies ? ------- music, reading, ------- Other physical puzzles ? ------- Big Rubik's cube fan. Also have various sequential movement puzzles (5x5x5 and Square-1) ------- Other programming projects ? ------- Only at work.... ------- Are you willing to show the group your source code (after removing your secret functions) ? ------- sure ------- Any other stuff you like to add ? ------- sugar ------- Do you know any other interesting problems ? ------- why do my socks keep disappearing in the washing machine? ------- What's the meaning of life ? ------- Be excellent to each other _______________________________________________________________________________ _______________________________________________________________________________ Message: 2 Date: Tue, 6 Jul 1999 11:27:20 +0100 From: "Ronald Stewart" Subject: Re: Parity OK, I think I understand this now: Each piece has a parity of +2, +4, or +6. This parity can be made negative by placing it on more "white" squares than "black" squares. In total, the parity of the pieces must sum to 0, otherwise not all the pieces can be covered (there will either be spare "white" or "black" squares). Thus, if the total parity reaches a number higher than the total remaining parity, then the puzzle cannot be completed and pieces must be removed. Am I right? - Ron -----Original Message----- From: Angus Walker To: eternity@onelist.com Date: Tuesday, July 06, 1999 11:08 AM Subject: [eternity] Parity >From: Angus Walker > >Christophe Weibel, thank you - that was a very clear explanation and I >understand it now. At what point in filling the board is parity able to >help dismiss an attempt? It would seem to be fairly late on unless >you've really built up a parity imbalance. At least I'll be able to >incorporate parity in my program now and look really sophisticated. >-- >Angus Walker _______________________________________________________________________________ _______________________________________________________________________________ Message: 3 Date: Tue, 06 Jul 1999 12:29:30 GMT From: David Clark Subject: Eternity hints Has anyone posted the 4 'hints' that are available for the eternity puzzle? (apart from the piece 34 hint) I have purchased the Eternity puzzle itself, but am not allowed to shell out money for the other puzzles. Given the chance of actually making a return on my investment (estimated return being about £30.00/10^100), my wife is arguing from a position of strength when she uses phrases like 'money down the drain' and 'eternity widow'.... So, please, could someone post the hints, or send them direct to me at eternity209@hotmail.com. (nice email address, eh? My normal address was being overrun by this mailing list....) Thanks, Dave _______________________________________________________________________________ _______________________________________________________________________________ Message: 4 Date: Tue, 6 Jul 1999 05:43:42 -0700 (PDT) From: =?iso-8859-1?q?Richard=20Heath?= Subject: Re: So, who are you guys, anyway ? > QUESTIONS > -------------- > What's your..... > > name :Richard Heath > age :23 > occupation :Student > city :York > country :UK > email :r_heath1@yahoo.com > homepage : > > ------- > Are you an active problem-solver, or just here out > of general interest ? > > ------- Active > ------- > Do you hope that by listening to this list, you'll > get the vital hint > that'll make you win the prize ? > ------- I hope I will get hints on writing a better program. > ------- > Do you own an Eternity puzzle ? > ------- Yup > ------- > What's your best attempt by hand ? > ------- Not many! > ------- > What's your best attempt by computer ? > ------- Into the 160's (If anyone is interested I will post my top 5 attempts on my web page) > ------- > What language / system / pc are you using ? > ------- C / Windows / P200 > ------- > What algorithms / heuristics do you use for Eternity > (brute force, backtracking, genetic algothims, > simulated annealing, > SECRET) ? > ------- Brute force variations > ------- > Did you type in all 209 pieces yourself ? > ------- Yes! > ------- > Did you try any simpler variations of Eternity ? > ------- Some of the smaller polydude problems on mathpuzzle.com > ------- > Do you know some other interesting web-sites ? > ------- No > ------- > Other hobbies ? > ------- Rock climbing, squash, walking, etc > ------- > Other physical puzzles ? > ------- Where have all of my biros gone? > ------- > Other programming projects ? > ------- Polymer simulations > ------- > Are you willing to show the group your source code > (after removing your > secret functions) ? > ------- Yes > ------- > Any other stuff you like to add ? > ------- Currently searching including against the grain placements. Only put the pieces in on Sunday so the code is being optimised quite a lot at the moment > ------- > Do you know any other interesting problems ? > ------- No > ------- > What's the meaning of life ? > ------- > Erm? I am not quite sure! _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at http://mail.yahoo.com _______________________________________________________________________________ _______________________________________________________________________________ Message: 5 Date: Tue, 06 Jul 1999 14:53:39 GMT From: David Clark Subject: Piece 34 starter position Hi fellow eternites. FYI, find attached a bitmap (zipped) of the starting piece 34 hint. Does this knowledge actually help? Probably not much... Regards, Dave ps. If anyone has the other 4 piece hints, I would like to receive them... thanks. ______________________________________________________ Get Your Private, Free Email at http://www.hotmail.com _______________________________________________________________________________ _______________________________________________________________________________ Message: 6 Date: Tue, 6 Jul 1999 09:53:30 -0500 From: "Richard Burchill" Subject: Re: who are you? [This message contained attachments] _______________________________________________________________________________ _______________________________________________________________________________ Message: 7 Date: Wed, 7 Jul 1999 01:14:19 +1000 From: "Mark Pursey" Subject: Re: Piece 34 starter position Thanks Dave! I was starting to think no one was ever going to post that, and didn't want to keep asking over and over. I must say i was kind of hoping it would be an edge piece ( My solver can't handle disconnected regions at the moment. Mark ----- Original Message ----- From: David Clark To: Sent: Wednesday, 7 July 1999 00:53 Subject: [eternity] Piece 34 starter position > > Hi fellow eternites. > > FYI, find attached a bitmap (zipped) of > the starting piece 34 hint. > Does this knowledge actually help? Probably not much... > > Regards, > Dave > > ps. If anyone has the other 4 piece hints, I would like to > receive them... thanks. > > > ______________________________________________________ > Get Your Private, Free Email at http://www.hotmail.com _______________________________________________________________________________ _______________________________________________________________________________ Message: 8 Date: Tue, 6 Jul 1999 16:20:28 +0100 From: cphipps@synetics.co.uk Subject: Re: Piece 34 starter position Chris Phipps@SYNETICS UK 07/06/99 04:20 PM Piece #34 is given away with the Eternity puzzle. Solving Meteor (10-pieces) gives you piece #187 (also in the middle on nowhere). Solving Delta (14-pieces) gives you another two pieces. Solving Heart (20-pieces) gives you two more. Thus it's possible to get the correct position of six pieces for the Eternity puzzle. Since I'm new too, here's my info... name : Chris Phipps age : 24 occupation : programmer city : London country : England e-mail : cphipps@synetics.co.uk ------- Are you an active problem-solver, or just here out of general interest ? ------- Active solver when I get the time. Here out of interest too though. ------- Do you hope that by listening to this list, you'll get the vital hint that'll make you win the prize ? ------- I'd like to think so, although I'd like to know if there's anyone else out there who's already obsessed with it. ------- Do you own an Eternity puzzle ? ------- Yes. I got one on the first day. I've bought the other three puzzles too. What a shame Toys'R'Us got the price structure of the puzzles wrong on the first day. ------- What's your best attempt by hand ? ------- Ooooh.... ------- What's your best attempt by computer ? ------- Ooooh.... ------- What language / system / pc are you using ? ------- C, C++ and other. P90, PII 350Mhz and seven Alpha workstations. ------- What algorithms / heuristics do you use for Eternity (brute force, backtracking, genetic algothims, simulated annealing, SECRET) ? ------- Brute force, custom genetic algorithm and another. ------- Did you type in all 209 pieces yourself ? ------- Yes. ------- Did you try any simpler variations of Eternity ? ------- Yes. Completed Meteor in about an hour although I'm stuck on Delta and Heart. ------- Do you know some other interesting web-sites ? ------- Nope. ------- Other hobbies ? ------- Badminton, cycling, swimming, reading, computing, maths puzzles. ------- Other physical puzzles ? ------- None, although I play a lot of computer puzzle games and strategy games. ------- Other programming projects ? ------- A few. Developing a couple of other puzzle solvers, including one that's nearly complete for Nonograms (or whatever they're called now) that they print in the Sunday Telegraph. ------- Are you willing to show the group your source code (after removing your secret functions) ? ------- Possibly, although most of it can be found in books. ------- Any other stuff you like to add ? ------- Anyone found a decent way of solving Heart? ------- Do you know any other interesting problems ? ------- TSP (GA). :-) ------- What's the meaning of life ? ------- 42. Cheers Douglas. Seriously, wish I knew. _______________________________________________________________________________ _______________________________________________________________________________ Message: 9 Date: Tue, 6 Jul 1999 12:05:53 -0500 From: "Richard Burchill" Subject: Start a programming thread. [This message contained attachments] _______________________________________________________________________________ _______________________________________________________________________________ Message: 10 Date: Tue, 6 Jul 1999 17:57:53 +0100 From: "Ronald Stewart" Subject: Re: Piece 34 starter position Thanks David... >From this piece alone, I have deduced that a piece with a right angle must fit into the right-angled gap (to the top on the bitmap). How many such pieces are there. Obviously piece 34 must be discounted, since it has alreadt been placed. _______________________________________________________________________________ _______________________________________________________________________________ Message: 11 Date: Tue, 6 Jul 1999 14:27:13 -0400 From: falk.werner@zurichre.com Subject: Nice try... Hi, I thought about the problem for several hours know, and I thought I had a nice initial approach: What we know for sure are the borders of the grid. Let's have a closer look at the 6 sides which are made up of half-triangles facing the border with their side of medium length. I scanned through the pieces and noticed that about 90 (92, with my manual investigation last night at 2am) may contribute to these 6 sides in either place, adding an average of 1.5 half-triangles to these sides with a length of 8 half-triangles. Just about an additional 90 (I counted 95) may contribute to these sides also, but due to their shape, they fit only to the ends/corners. Let's say you need 5 four of the ones you can fit anywhere along that sideline to start some maths, regardless of the corner-ones only complicating it even further. Picking 5 out of 90 (sequence doesn't matter) gets us to about 5 billion combinations. Let's assume that a nice program will try to fit each of these sets of 5 together to fit along the sideline in 1 second. This leads to close to 170 years of calculation to find out which combinations actually work. Now we need six of these sets to fill the borders. That gets us to even more combinations. The numbers above are not truly correct, because you have to actually pick the six sets all ut once out of the 90, so that reduces the possibilities. On the other hand, if you take the corner-ones into consideration as well, this is truly overcompensated. Anyway, let's say it will take 200 years to split the problem into "Reduce the overall grid by picking a set of borderlines" x "Solve Eternity with the remaining about 180 pieces". However, keep in mind that many of the combinations would not lead to a 'valid' shape facing the rest of the grid. Okay, nice thought, nice try, bad maths, but if we had more approaches like this.... You never know. Falk _______________________________________________________________________________ _______________________________________________________________________________ Message: 12 Date: Tue, 6 Jul 1999 14:39:47 -0400 From: falk.werner@zurichre.com Subject: Another one... Another approach: What do you usually do when you solve a puzzle and you have a piece that doesn't fit any more? You try to exchange it with another piece already on the set. Let's say you filled the grid with 160 pieces, and there is no way to get any further, because the remaining pieces don't fit. One possibility is to remove the pieces you just placed on the board, okay. Another one is to make up pairs out of the remaining ones and check, if you could combine them in a way that the final shape equals another pair of pieces already on the grid. I don't know how likely it is to find one, but in case you are lucky, you can exchange these pairs and have two new 'remaining pieces' of different shapes. In other words: Once you have a partial potential solution, you may want to take advantage also of the pieces you dropped on the board already. You don't necessarily have to backtrack by removing pieces again. To me it sounds like a more random/brute force logic in the beginning to fill the grid as much as possible, and then use more intelligent trade-like approaches. Any thoughts on that one? Falk _______________________________________________________________________________ _______________________________________________________________________________ Message: 13 Date: Tue, 6 Jul 1999 14:55:25 -0400 From: falk.werner@zurichre.com Subject: And a third one... Many of the pieces have not identical (I know) but similar shapes. Look at possible combinations of full triangles: 209 pieces carry two triangles aligned With 3 triangles the shape comes in 3 different shapes: all next to each other, fish shape symmetrical and fish shape with tail slightly bended. With 4 triangles, we have 9 different shapes occurring in the pieces. With 5 triangles, we have 3 different shapes. The overall approach is: Place the full triangles first, and then decide later which piece you actually want to choose. Only 178 out of 1254 triangles (full or half, like at the borders) are made up out of half triangles. My robust approach is (still foggy, though): Place the first 209 triangles randomly across the board, checking that the surroundings allow enough space to finally expand it to an area of 6 triangles (full and half ones). For all the following steps, in case you are running into trouble because there is not enough free space left any more, you are allowed to move a triangle if it works then, or to exchange/regroup the clusters. Add another triangle to each of these (209 times). Add another triangle (209 times, but be aware of the different shapes: tetragon, straight and bended fish as well as the surroundings). Add another triangle where you have space (183 times, be aware of the different shapes again, and the surrounding space left, of course). Add another triangle where you have space (68 times, rules as above again). Again, whenever you run into trouble, try the 'move' and 'regroup' approach. Finally try to fill in the half triangles and select a dedicated shape. If you managed to get that far, you still have a choice of about 5 different shapes for each piece. Doesn't sound too bad, does it? Needs a sophisticated program(mer), though. Think about it and let me know about your thoughts. By the way, I'm not a programmer, and there is no way I will win that puzzle anyway. I would be proud already if anybody of our group manages to find a solution. Falk _______________________________________________________________________________ _______________________________________________________________________________ Message: 14 Date: Tue, 6 Jul 1999 15:00:21 -0400 From: falk.werner@zurichre.com Subject: By the way: Re: Who are You ? QUESTIONS -------------- What's your..... name :Falk WERNER age :33 occupation :Manager in Consulting city :Nationality: German Citizenship: Stuttgart(GER) Residence: New Jersey(US), Current location: Zurich(CH) country :see above email :at the client's site: falk.werner@zurichre.com homepage :None ------- How did you hear about this list (website / friends)? ------- Accidentally running across the topic while searching onn the Internet for an article from Ian Stewart, mentioned in a book about statistics and its daily use ("Der Hund, der Eier legt. Erkennen von Fehlinformationen durch Querdenken.", rororo Verlag, ist ganz witzig). Finally ended on www.mathpuzzle.com with the link to ONEList. ------- Are you an active problem-solver, or just here out of general interest ? ------- Kind of both. I was thinking about different approaches than basically searching through more or less the whole tree. ------- Do you hope that by listening to this list, you'll get the vital hint that'll make you win the prize ? ------- No. Well, maybe a vital hint, but I am not a programmer, so somebody else will be faster anyway. ------- Do you own an Eternity puzzle ? ------- No. ------- What's your best attempt by hand ? ------- N/A. ------- What's your best attempt by computer ? ------- In the 160's with the ScreenSolver. ------- What language / system / pc are you using ? ------- At the moment about around 5kg of what is mostly water, with some millions of lightnings per seconds. ------- What algorithms / heuristics do you use for Eternity (brute force, backtracking, genetic algothims, simulated annealing, SECRET) ? ------- I think only a 'SECRET' would do it, but I don't have one yet, if there is any at all. Otherwise, the only secret is 'luck', because the calculations of all the people in the list states rather clearly that there is no solution to be found in reasonable time without a good portion of luck. ------- Did you type in all 209 pieces yourself ? ------- No. ------- Did you try any simpler variations of Eternity ? ------- No. ------- Do you know some other interesting web-sites ? ------- No time, unfortunately... Just looking at the stocks. ------- Other hobbies ? ------- Skiing, Windsurfing, basically any type of sport, clubbing, Alfa Romeos. About 15 years ago, I spent afternoons and evenings with a friend trying to solve math- and jigsawpuzzles by M. Gardner and others. ------- Other physical puzzles ? ------- The male/female one. ------- Other programming projects ? ------- No. ------- Are you willing to show the group your source code (after removing your secret functions) ? ------- I doubt that I'm able to produce one. ------- Any other stuff you like to add ? ------- If you look at all the puzzles from Martin Gardner, it is most important not to start implementing a problem solver right away, but think about the problem for at least a week without actually doing something. I think a more "natural" approach will do better. ------- Do you know any other interesting problems ? ------- How do we get rid of all the people bothering us with silly questions about 'Eternity', Leaving the few which are actually working on the problem in peace? ------- What's the meaning of life ? ------- As with eternity, out of zillions of possible combinations of amino acids, more by chance everything evolved. Feel lucky, have fun and enjoy it. ------- How are you going to spend your 1000.000 ? ------- Invest it and work only 50% of the regular 40 hrs., instead of 120%+. _______________________________________________________________________________ _______________________________________________________________________________ Message: 15 Date: Tue, 6 Jul 1999 23:06:03 EDT From: MasterMB@aol.com Subject: a quick (maybe) question I've seen reference to the 'fact' that each of the 209 pieces have twelve orientations. Is this the result of the ability to rotate each piece 30 degrees (* 12), OR is the result of the ability to rotate each piece 60 degrees (* 6) AND to flip each piece over and do the same with its mirror image? Any ideas? Thanks, Marc Becker _______________________________________________________________________________ _______________________________________________________________________________ Message: 16 Date: 7 Jul 1999 04:43:30 -0000 From: Dsaund2773@aol.com Subject: Re: Marc Becker and orientation issue Marc, If you take a hexagon made up of triangles and bisect these triangles, you have 12 segments. Twelve possible orientations. If you flip this hexagon over you have another 12. But these two hexagons are mirror images so you have 12 orientations. If a piece does not match up with the grid below, it is "against the grain". Then what? I'm not sure. If you rotate the hexagon 30 degrees, then what? I'm not sure. I am trying to number the twelve orientations with the first 12 numbers so that up/down and east/west parity are maintained. If you got the answer please Email. How many different ways can you number the twelve orientations using just the first 12 numbers? 39,916,800. This is N!/N. Trying to solve Eternity may show us new approaches to other things. 1 million pounds well spent. Dick Saunders _______________________________________________________________________________ _______________________________________________________________________________ Message: 17 Date: Wed, 7 Jul 1999 16:52:04 +1000 From: "Mark Pursey" Subject: Re: a quick (maybe) question It seems generally accepted that each piece must be rotated 60 degrees at a time, so that a triangle 'long' edge always aligns with the grid on the board. Thus the 12 comes from the 6 rotations * 2 for flipping. Without the grid marked on the pieces as (wisely) recommended, i think it may be easy to start using 30 degree rotations and waste a lot of time trying to make them fit, If eternity could be solved using a mixture of these orientations, you'd probably have a piece or 2 left over, and that i'm sure would be extremely frustrating. Mark ----- Original Message ----- From: To: Sent: Wednesday, 7 July 1999 13:06 Subject: [eternity] a quick (maybe) question > From: MasterMB@aol.com > > I've seen reference to the 'fact' that each of the 209 pieces have twelve > orientations. Is this the result of the ability to rotate each piece 30 > degrees (* 12), OR is the result of the ability to rotate each piece 60 > degrees (* 6) AND to flip each piece over and do the same with its mirror > image? Any ideas? > > Thanks, > Marc Becker > > --------------------------- ONElist Sponsor ---------------------------- > > How has ONElist changed your life? > Share your story with us at http://www.onelist.com > > ---------------------------------------------------------- -------------- > _______________________________________________________________________________ _______________________________________________________________________________ Message: 18 Date: Wed, 7 Jul 1999 02:46:09 -0500 From: path@multipro.com.au Subject: quick (maybe) question and an 'off-grid' exercise Patrick M Hamlyn@MULTIPROGRAMMING 07/07/99 03:46 PM From: "Mark Pursey" >It seems generally accepted that each piece must be rotated >60 degrees at a time, so that a triangle 'long' edge always >aligns with the grid on the board. Thus the 12 comes from >the 6 rotations * 2 for flipping. Without the grid marked on >the pieces as (wisely) recommended, i think it may be easy >to start using 30 degree rotations and waste a lot of time >trying to make them fit, If eternity could be solved using a >mixture of these orientations, you'd probably have a piece >or 2 left over, and that i'm sure would be extremely >frustrating. I don't believe you can place any pieces at 30 degrees. Try placing just one along the edge rotated 30 degrees from a 'legal orientation'. If you place so much as one piece that way, I believe you will have to place them all that way to get them to fit together. Then you'll have to rotate the grid 30 degrees too of course. Here's an exercise on off-grid drafters you can try: Draw a grid, oriented 'right way up', ie with triangle sides horizontal and 60 degrees either side of vertical. Now draw all the lines which bisect each triangle (3 lines per triangle). You should have a grid with 6 'mini-drafters' per triangle, and because all the horizontal and vertical lines are now marked, you can easily trace 'rectilinear borders' around areas which can contain off-grid pieces. Also, rotating the grid 60 degrees gives you another set of horizontal and vertical lines. Now if you tile such a rectilinear region with Eternity pieces, you should be able to easily see that it can be shifted one half triangle-side right or left, and the pieces in it will then be 'off-grid'. Also note that if you tile a rectangle with a width which is not a whole number of triangle-sides, or a height which is an odd number of triangle-heights, and place it 'off-grid', it can be flipped 'in place' so that the pieces are back on-grid. Now draw an outline around a shape, using only lines which are vertical, or 60 degrees either side of horizontal (eg an equilateral triangle with side lengths which are a multiple of the height of a base triangle). Make this shape big enough to hold say 10 pieces exactly. You'll notice that such a shape can be placed 'off-grid' by shifting it half a triangle-height up or down. Now work out whether it is possible to place pieces off-grid in an Eternity solution using this technique. If you can't work out what I mean, take a look at the hexadudes on Ed Pegg's site. The triangular piece (number 7 if you number them in rows) is an example of a non-rectilinear grid-ambiguous piece. You can mark the grid on this piece in two ways - with a triangle in the middle and three drafters around it, or with 6 drafters all sharing a vertex in the centre. The exercise is to work out whether this means that you can place the unrotated piece on the grid in two different sets of positions without mucking up the placing of the rest of the pieces. _______________________________________________________________________________ _______________________________________________________________________________ Message: 19 Date: Wed, 7 Jul 1999 03:04:21 -0500 From: path@multipro.com.au Subject: Re: quick (maybe) question and an 'off-grid' exercise Patrick M Hamlyn@MULTIPROGRAMMING 07/07/99 04:04 PM Oops Ed moved the pieces around. This is now piece 10 numbering them in rows. >take a look at the hexadudes on Ed >Pegg's site. The triangular piece >(number 7 if you number them in rows) _______________________________________________________________________________ _______________________________________________________________________________ Message: 20 Date: Wed, 07 Jul 1999 10:10:43 +0200 From: Andreas Gammel Subject: status + complexity hi, - 190 members (woops, there's another one, 191, o no darn, its an unsubsription, 189)! That's more than the current best solution. Before long, each member can be assigned a single Eternity piece and decide where it goes on the board... - we now have members from the UK, USA, Italy, Holland, Switzerland, Sweden, Denmark, France, Australia, Gernamy, Chezoslovakia, Belgium, Spain, Canada, South Africa, Turkey, Argentina and lots from .org's .com's and net's - we had 163 messages so far, sent by around 30 people - I received about 50 C.V.'s. Some nice answers about cats running over partial solutions, wives forbidding any more useless money spending, screensavers never doing any better than 155, Anybody dream about Eternity yet ? Well, I did last night. Happens to me a lot, when I think about certain subjects too long. A little more crap about complexity: I wouldnt be surprised if the problem is never solved. It's the same as the Traveling Salesman for 100 cities, which has never been solved. But I do think some extra effort is needed to calculate the actual complexity of the problem. This is a difficult problem in itself. We assume that the order is 10^200 or the likes, but we don't actually know that. We do not know the 'dead branches' of the tree before we've actually been there. Another interesting thought is that 'mathematical' solutions, using operators like +,*,^, are accepted as 'real' solutions, whereas algorithms, using arrays and assigments, are only accepted when they actually produce a solution. Eg if we want to know the number of branches of Eternity than an answer like 2^209 would be completely acceptable, whereas a search algorithm is not, although in BOTH cases the actual number is never computed. for example: a = 1; for x = 1 to 100; a = a * 2; next x equals 2 ^ 100 a = 1; for x = 1 to 100; a = a * 2; a = a + 1; next x equals no known math expression Does this not implicate that 'mathematics' itself is limited, because it is memory-free ? Who knows the answer to such questions ? It is well known that Stephen Hawking was only able to express and solve his thoughts about singularities in black holes after Roger Penrose had developed a new type of math for him. Andreas _______________________________________________________________________________ _______________________________________________________________________________ Message: 21 Date: Wed, 7 Jul 1999 18:27:53 +1000 From: "Mark Pursey" Subject: the ambiguous hexadude I think i might be missing something with the ambiguity of the triangular hexadude. The only way i can see to make it an area of 6 drafters (ie a hexadude) is to use 6 drafters, with the vertex in the middle. In the case of one triangle surrounded by 3 drafters, the area doesn't add up. If a long side is 1 unit, then the side length of the 6 drafter triangle is root(3), whereas the side length of the second triangle is 3/2. ie the inner triangle is 30 deg off and has side length root(3)/2. But maybe i'm just visualizing it wrong, if someone has a picture of this alternative version i'd like to see it. Mark ----- Original Message ----- From: To: Sent: Wednesday, 7 July 1999 18:04 Subject: Re: [eternity] quick (maybe) question and an 'off-grid' exercise > From: path@multipro.com.au > > > > > > Patrick M Hamlyn@MULTIPROGRAMMING > 07/07/99 04:04 PM > > Oops Ed moved the pieces around. This is now piece 10 numbering them in > rows. > > >take a look at the hexadudes on Ed > >Pegg's site. The triangular piece > >(number 7 if you number them in rows) > > > > --------------------------- ONElist Sponsor ---------------------------- > > ONElist: your connection to people who share your interests. > > ---------------------------------------------------------- -------------- > _______________________________________________________________________________ _______________________________________________________________________________ Message: 22 Date: Wed, 7 Jul 1999 09:34:10 +0100 From: cphipps@synetics.co.uk Subject: Re: status + complexity Chris Phipps@SYNETICS UK 07/07/99 09:34 AM Andreas said: > for example: > > a = 1; for x = 1 to 100; a = a * 2; next x > equals > 2 ^ 100 > > a = 1; for x = 1 to 100; a = a * 2; a = a + 1; next x > equals > no known math expression I think 2^(x+1) - 1 will work. ie. (2^101) - 1 _______________________________________________________________________________ _______________________________________________________________________________ Message: 23 Date: Wed, 07 Jul 1999 10:55:47 +0200 From: Andreas Gammel Subject: Re: status + complexity cphipps@synetics.co.uk wrote: > > a = 1; for x = 1 to 100; a = a * 2; a = a + 1; next x > > equals > > no known math expression > > I think 2^(x+1) - 1 will work. > ie. (2^101) - 1 ok, bad example, but you get the point, right ? _______________________________________________________________________________ _______________________________________________________________________________ Message: 24 Date: Wed, 7 Jul 1999 11:18:17 +0200 From: Chris McLaren Subject: Re: Welcome to eternity@onelist.com QUESTIONS -------------- What's your..... name : Chris McLaren age : 49 occupation : Mineralogist city : Randburg country : South Africa email : chrism@mintek.co.za homepage : ------- How did you hear about this list (website / friends) ? ------- My son (s9460462@vagus.up.ac.za ) came across an article in a local magazine and then referred me to mathpuzzle.com ------- Are you an active problem-solver, or just here out of general interest ? ------- Out of interest (my son is active - I'll get him to fill in his own questionnaire once he gets back to University) ------- Do you hope that by listening to this list, you'll get the vital hint that'll make you win the prize ? ------- Who knows? ------- Do you own an Eternity puzzle ? ------- No, but we have made a set from the available info ------- What's your best attempt by hand ? ------- 6 for starters! ------- What's your best attempt by computer ? ------- 161 - Screen Solver ------- What language / system / pc are you using ? ------- Win95, Pentium II - 400 ------- What algorithms / heuristics do you use for Eternity (brute force, backtracking, genetic algothims, simulated annealing, SECRET) ? ------- Not applicable ------- Did you type in all 209 pieces yourself ? ------- N/A ------- Did you try any simpler variations of Eternity ? ------- No , but sets are on order ------- Do you know some other interesting web-sites ? ------- Lots ------- Other hobbies ? ------- Playing bass guitar and listening to music of the Beatles, Stones, Procol Harum, Dave Clark 5, Deep Purple, Uriah Heep, Led Zep, DDDBM&T, etc. ------- Other physical puzzles ? ------- ex-Rubik's Cube and Tangle ------- Other programming projects ? ------- None currently ------- Are you willing to show the group your source code (after removing your secret functions) ? ------- N/A ------- Any other stuff you like to add ? ------- Not at this stage ------- Do you know any other interesting problems ? ------- Counting the total number of triangles in hexagons, heptagons, etc. to find a general formula to predict the total (Anyone got one?) ------- What's the meaning of life ? ------- To live? ------- How are you going to spend your 1000.000 ? ------- If we know we're about to submit the first correct entry, we'll spend some front-up and get tickets to fly to the UK for hand delivery of our entry! -----Original Message----- From: eternity-owner@onelist.com [SMTP:eternity-owner@onelist.com] Sent: Monday, July 05, 1999 2:38 PM To: chrism@mintek.co.za Subject: [eternity] Welcome to eternity@onelist.com Hello, Welcome to the list. Please take a moment to review this message............ Andreas Gammel, List Owner _______________________________________________________________________________ _______________________________________________________________________________ Message: 25 Date: Wed, 7 Jul 1999 10:18:21 +0100 From: "Ronald Stewart" Subject: Re: status + complexity >> a = 1; for x = 1 to 100; a = a * 2; a = a + 1; next x >> equals >> no known math expression > >I think 2^(x+1) - 1 will work. >ie. (2^101) - 1 most certainly agreed. x = 1: a = 2 + 1 x = 2: a = 2^2 + 2 + 1 x = 3: a = 2^3 + 2^2 + 2 + 1 ... x = 100: a = 2^100 +... +2+1 = 2^101 - 1 _______________________________________________________________________________ _______________________________________________________________________________