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You are not logged in. #1 20110820 21:10:11
Museum Design Puzzle  Help out the squareheaded architectThis puzzle features a truly squareheaded architect who wants to design a museum in the shape of an nxnxh square box, where n is an integer denoting the side length and h is the height of the building. Last edited by Fausto Morales (20110822 01:35:54) #2 20110820 21:50:15
Re: Museum Design Puzzle  Help out the squareheaded architectHi Fausto Morales; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #3 20110820 22:16:25
Re: Museum Design Puzzle  Help out the squareheaded architectHi! Last edited by Fausto Morales (20110820 22:32:08) #4 20110820 22:43:57
Re: Museum Design Puzzle  Help out the squareheaded architectHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20110820 23:00:04
Re: Museum Design Puzzle  Help out the squareheaded architectHi, Last edited by Fausto Morales (20110820 23:16:20) #6 20110820 23:14:09
Re: Museum Design Puzzle  Help out the squareheaded architectHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20110820 23:22:26
Re: Museum Design Puzzle  Help out the squareheaded architectAbsolutely, I also expect some other values of n to fail besides n=5 (If indeed n=5 turns out to fail  I haven't checked this by computer nor have I been able to prove that it fails.) However, n=6 works and I expect infinitely many other values of n to work out too. Last edited by Fausto Morales (20110821 01:52:13) #8 20110820 23:35:33
Re: Museum Design Puzzle  Help out the squareheaded architectOkay, if I get anything meaningful I will post it. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #9 20110821 02:40:51
Re: Museum Design Puzzle  Help out the squareheaded architectHINT: Last edited by Fausto Morales (20110821 11:00:52) #10 20110821 18:52:59
Re: Museum Design Puzzle  Help out the squareheaded architectYou can find the 26 symmetric museum designs that we have found so far.What is amazing to me is that such a set of basic, uniform, regular design conditions  symmetrically splitting an nxn square into n polygons made of 1x1 squares, all polygons having the same area, perimeter and number of neighbors  appears to impose the presence of some fairly irregular, zigzagging shapes already for n=8. If this is any indication, designs will get increasingly interesting and original as n grows. P.S.: This impression is further supported by the symmetric designs for n=9 that I've just found  as you can see at the "here" link, these also have a few really twisted rooms. Is this going to be the fun trend as n grows? The next big challenge ahead is to go for the n=10 record. HINTS: 1. Build connected, planar (no crossings between edges), 4regular graphs and study their suitability to represent connections of 10polyominoes of the same perimeter symmetrically placed within a 10x10 square. Keep in mind that the same graph may have several potentially useful planar embeddings (i.e. "presentantions" on the plane). 2. A trick to get you started with the graphs: a. Put 4 vertices in a square configuration and join them with edges to form a square. b. Similarly, use 6 more vertices to place a hexagon within the square. c. Draw 2 more edges between vertices of the hexagon (you have some choices here). d. Complete the graph by connecting some vertices of the square with some vertices of the hexagon, so that when you are done each vertex is connected to 4 others and there are no crossings between edges. e. See if you can use your graph to help you produce a correct partition of the 10x10 square into 10polyominoes. If not, repeat steps (c) and (d) and try again. Have lots of fun! Last edited by Fausto Morales (20110825 21:43:34) #11 20110826 17:43:38
Re: Museum Design Puzzle  Help out the squareheaded architectOh, in case you were wondering: at least 1 solution exists to the 10x10 design. I will add it to the collection on September 22, unless someone can find it first... Last edited by Fausto Morales (20110903 17:20:03) #12 20110830 17:58:55
Re: Museum Design Puzzle  Help out the squareheaded architectLast edited by namealreadychosen (20110830 18:00:22) #13 20110830 18:40:46
Re: Museum Design Puzzle  Help out the squareheaded architect1. Seems that a row is missing in your 10x10 design. Couldn't fill in the gap with equal perimeters and symmetry simultaneously. Please repost. Last edited by Fausto Morales (20110830 18:53:31) #14 20110831 16:42:53
Re: Museum Design Puzzle  Help out the squareheaded architect1. Sorry, the10x10 was wrong. #15 20110904 22:37:34
Re: Museum Design Puzzle  Help out the squareheaded architectYeah, that 9x10 10x10 square REALLY threw me off for a minute! It took a second to realize we were missing a few spots there.. #16 20110905 20:23:34
Re: Museum Design Puzzle  Help out the squareheaded architectHints for the 2 symmetric solutions known so far to the 10x10 design: 