[math-fun] Magic cubes
I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R.
MathWorld Headline News Perfect Magic Cube of Order 5 Discovered By Eric W. Weisstein November 18, 2003--This week, German mathematics teacher Walter Trump and French software engineer Christian Boyer announced the discovery of a perfect magic cube of order 5, thus settling the long-standing question of the existence of such a cube. http://mathworld.wolfram.com/news/2003-11-18/magiccube/ Richard Guy <rkg@cpsc.ucalgary.ca> wrote: I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Thanks! I shd have been more careful. What is a ``perfect'' magic cube? Trenkler's specimens have the 3n^2 ``rows'' and the 4 body diags magic. Is it possible to get (any of) the 6n ``two- dimensional'' (perhaps the 12 ``visible'', ``face'') diagonals magic? R. On Fri, 7 Mar 2008, Ed Pegg Jr wrote:
MathWorld Headline News Perfect Magic Cube of Order 5 Discovered By Eric W. Weisstein November 18, 2003--This week, German mathematics teacher Walter Trump and French software engineer Christian Boyer announced the discovery of a perfect magic cube of order 5, thus settling the long-standing question of the existence of such a cube. http://mathworld.wolfram.com/news/2003-11-18/magiccube/
Richard Guy <rkg@cpsc.ucalgary.ca> wrote: I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R.
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Trenkler must be using a different definition of Magic Cube. [Some older books only require the magic sum for the exterior face diagonals, etc.] Martin Gardner published my proof of non-existance of a magic 4^3 in Scientific American c. 1974. The non-existance of 2^3 is trivial, and non-existance of 3^3 is easy. The 5^3 must have its average value in the center, similar to the 3^2. This prevents 5^4. Trump & Boyer constructed 6^3 and then 5^3 a few years ago, using a mix of cleverness & computer search. 7^3 and larger is apparently very old news. Boyer has a web page about the 5^3 and 6^3; I think it's multimagie.com, and covers a lot of different kinds of magic objects. He's on Math-Fun, so he'll likely contact you directly. I've worked on showing non-existance of 6^4 from time to time, without success. I have a complicated sum-restriction for 6^3, but it (obviously) doesn't preclude them. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of Ed Pegg Jr [ed@mathpuzzle.com] Sent: Friday, March 07, 2008 9:41 AM To: math-fun Subject: Re: [math-fun] Magic cubes MathWorld Headline News Perfect Magic Cube of Order 5 Discovered By Eric W. Weisstein November 18, 2003--This week, German mathematics teacher Walter Trump and French software engineer Christian Boyer announced the discovery of a perfect magic cube of order 5, thus settling the long-standing question of the existence of such a cube. http://mathworld.wolfram.com/news/2003-11-18/magiccube/ Richard Guy <rkg@cpsc.ucalgary.ca> wrote: I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The direct access to this page is www.multimagie.com/English/Perfectcubes.htm Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Schroeppel, Richard Envoyé : vendredi 7 mars 2008 18:06 À : math-fun Objet : Re: [math-fun] Magic cubes Trenkler must be using a different definition of Magic Cube. [Some older books only require the magic sum for the exterior face diagonals, etc.] Martin Gardner published my proof of non-existance of a magic 4^3 in Scientific American c. 1974. The non-existance of 2^3 is trivial, and non-existance of 3^3 is easy. The 5^3 must have its average value in the center, similar to the 3^2. This prevents 5^4. Trump & Boyer constructed 6^3 and then 5^3 a few years ago, using a mix of cleverness & computer search. 7^3 and larger is apparently very old news. Boyer has a web page about the 5^3 and 6^3; I think it's multimagie.com, and covers a lot of different kinds of magic objects. He's on Math-Fun, so he'll likely contact you directly. I've worked on showing non-existance of 6^4 from time to time, without success. I have a complicated sum-restriction for 6^3, but it (obviously) doesn't preclude them. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of Ed Pegg Jr [ed@mathpuzzle.com] Sent: Friday, March 07, 2008 9:41 AM To: math-fun Subject: Re: [math-fun] Magic cubes MathWorld Headline News Perfect Magic Cube of Order 5 Discovered By Eric W. Weisstein November 18, 2003--This week, German mathematics teacher Walter Trump and French software engineer Christian Boyer announced the discovery of a perfect magic cube of order 5, thus settling the long-standing question of the existence of such a cube. http://mathworld.wolfram.com/news/2003-11-18/magiccube/ Richard Guy <rkg@cpsc.ucalgary.ca> wrote: I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Richard, supplemental information. If you plan to update problem D15 of UPINT, read my paper published in The Mathematical Intelligencer, Spring 2005, pages 52-64: I organized this paper around nine quotations from your own text, UPINT 3rd edition 2004. The 5x5x5 magic cube question is one of them (quotation #8, pages 60-62 of the M.I. paper). Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Richard Guy Envoyé : vendredi 7 mars 2008 17:36 À : Math Fun Objet : [math-fun] Magic cubes I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Thankyou, Christian! R, Could you, or someone else, clarify the definition of `magic cube'? For magic squares, there are (a) n rows & n columns (b) diagonals, 2 main & 2(n-1)broken. I understand that, in a magic square, (a) & main (b) have the magic total, and that a `pandiagonal' (`Nasik'?) square has (a) & all of (b). When we come to cubes, the situation is more complicated. There are (a) n^2 rows in each of 3 orthogonal directions (b) `2-dimensional' diagonals in 6 directions, which may be classified as [don't think I've got the numbers right]: (b1) 12 `face' (`visible') diagonals, (b2) 6(n-2) `edge' (`interior') diags, (b3) 12(n-1) broken `face' diagonals, (b4) 12(n-2)^2? broken interior diags. (c) `3-D' diags in 4 directions: (c1) 4 main body diagonals, (c2) 12(n-2)? with ends on edges, (c3) 12(n-2)^2? with ends on faces [these last two being broken]. Trenkler's cubes have (a) and (c1). Does a `perfect' cube have (a), (c1) & (b1) ?? On Mon, 10 Mar 2008, Christian Boyer wrote:
Richard, supplemental information.
If you plan to update problem D15 of UPINT, read my paper published in The Mathematical Intelligencer, Spring 2005, pages 52-64: I organized this paper around nine quotations from your own text, UPINT 3rd edition 2004. The 5x5x5 magic cube question is one of them (quotation #8, pages 60-62 of the M.I. paper).
Christian.
-----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Richard Guy Envoyé : vendredi 7 mars 2008 17:36 À : Math Fun Objet : [math-fun] Magic cubes
I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R.
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Richard, you are perfectly right. With cubes, the situation is more complicated. And with hypercubes, the situation is much more complicated! On your question, as mentioned in my MI paper, a nxnxn "perfect magic cube" has 3n²+6n+4 magic lines: your (a) + (c1) + (b1) + also (b2). With cubes, from each cell, you have 13 possible directions. The best possible cubes are often called "pandiagonal perfect magic cubes". See www.multimagie.com/English/Panperfectcubes.htm, or www.multimagie.com/English/PanperfectMMC.htm all is magic in any direction from any cell! It seems impossible to create a pandiagonal perfect magic cube of an order smaller than 8, but I have never seen such a proof. Tomorrow, I can send you more details on your question if you wish. Directly, details can be too long and boring for [math-fun]. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Richard Guy Envoyé : lundi 10 mars 2008 20:12 À : math-fun Objet : Re: [math-fun] Magic cubes Thankyou, Christian! R, Could you, or someone else, clarify the definition of `magic cube'? For magic squares, there are (a) n rows & n columns (b) diagonals, 2 main & 2(n-1)broken. I understand that, in a magic square, (a) & main (b) have the magic total, and that a `pandiagonal' (`Nasik'?) square has (a) & all of (b). When we come to cubes, the situation is more complicated. There are (a) n^2 rows in each of 3 orthogonal directions (b) `2-dimensional' diagonals in 6 directions, which may be classified as [don't think I've got the numbers right]: (b1) 12 `face' (`visible') diagonals, (b2) 6(n-2) `edge' (`interior') diags, (b3) 12(n-1) broken `face' diagonals, (b4) 12(n-2)^2? broken interior diags. (c) `3-D' diags in 4 directions: (c1) 4 main body diagonals, (c2) 12(n-2)? with ends on edges, (c3) 12(n-2)^2? with ends on faces [these last two being broken]. Trenkler's cubes have (a) and (c1). Does a `perfect' cube have (a), (c1) & (b1) ?? On Mon, 10 Mar 2008, Christian Boyer wrote:
Richard, supplemental information.
If you plan to update problem D15 of UPINT, read my paper published in The Mathematical Intelligencer, Spring 2005, pages 52-64: I organized this paper around nine quotations from your own text, UPINT 3rd edition 2004. The 5x5x5 magic cube question is one of them (quotation #8, pages 60-62 of the M.I. paper).
Christian.
-----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Richard Guy Envoyé : vendredi 7 mars 2008 17:36 À : Math Fun Objet : [math-fun] Magic cubes
I have a question in UPINT, D15: ``Has anyone constructed a 5 x 5 x 5 magic cube, or proved its impossibility'' (just possibly asked by Rich Schroeppel?). I've recently downloaded a 2-page article by Maria'n Trenkler, An algorithm for making magic cubes, which was evidently published in Pi Mu Epsilon J, 12 #2 (Spring 2005) 105-106. It produces magic cubes of any order > 2. Am I the only person not to know about this? R.
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participants (4)
-
Christian Boyer -
Ed Pegg Jr -
Richard Guy -
Schroeppel, Richard