魔術正方體：修订间差异

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2022年11月23日 (三) 10:11的最新版本

在数学中，魔術正方體指三維的幻方，也就是排列成n × n × n正方體的一組不重複整數，其中每行、每列、每個柱及四條空間對角線（英语：Space diagonals）上數字的和均相同，等於立方體的幻方常數，記為M₃(n)。^[1]若魔術立方體由數列1, 2, ..., n³構成，則可以證明其幻方常數為（OEIS數列A027441）

M_{3}(n)={\frac {n(n^{3}+1)}{2}}

另外，如果每個截面對角線上的數字之和亦等於幻方常數，則稱此立方體為完美魔方（英语：perfect magic cube）；否則，稱其為半完美魔方（英语：semiperfect magic cube）。數字n稱為魔方的階。如果幻方破碎空間對角線（英语：broken space diagonal）上的數字和也等於幻方常數，則稱其為泛對角線立方體。

參見

注釋

^ Weisstein, Eric W. (编). Magic Cube. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2016-12-04]. （原始内容存档于2021-03-07）（英语）.

外部連結

埃里克·韦斯坦因. Magic Cube. MathWorld.
Harvey Heinz, All about Magic Cubes （页面存档备份，存于互联网档案馆）
Marian Trenkler, Magic p-dimensional cubes （页面存档备份，存于互联网档案馆）
Marian Trenkler, An algorithm for making magic cubes （页面存档备份，存于互联网档案馆）
Marian Trenkler, On additive and multiplicative magic cubes
Ali Skalli's magic squares and magic cubes （页面存档备份，存于互联网档案馆）

[:0-1] Weisstein, Eric W. (编). Magic Cube. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2016-12-04]. （原始内容存档于2021-03-07）（英语）.

[1]

@@ 第1行： / 第1行： @@
+{{about|數學上的魔術立方體|益智玩具|魔術方塊}}
-{{about|the mathematical concept|the flashbulb cartridges|Magicube|the puzzle|Rubik's Cube}}
-[[File:Simple Magic Cube.svg|thumb|right|An example of a 3&nbsp;&times;&nbsp;3&nbsp;&times;&nbsp;3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a [[简易魔术正方体]].]]
+[[File:Simple Magic Cube.svg|thumb|right|一個3&nbsp;&times;&nbsp;3&nbsp;&times;&nbsp;3[[简易魔术正方体]]的例子]]
-以[[数学]]方面論述，'''魔術正方體''' 在[[維度]]上相當於[[幻方]]，也就是以''n''&nbsp;&times;&nbsp;''n'' 方式排列的方體，在每個線段交點填上任意不重複的[[整数]]，並使得每行、每列及每個柱上數字的和相同。而此立方體的[[幻方常數]]表示為''M''<sub>3</sub>(''n'').<ref name=":0">{{Cite web|url=http://mathworld.wolfram.com/MagicCube.html|title=Magic Cube|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2016-12-04}}</ref> It can be shown that if a magic cube consists of the numbers 1, 2, ..., ''n''<sup>3</sup>, then it has magic constant {{OEIS|id=A027441}}
+在[[数学]]中，'''魔術正方體'''指[[維度|三維]]的[[幻方]]，也就是排列成''n''&nbsp;&times;&nbsp;''n''&nbsp;&times;&nbsp;''n''正方體的一組不重複[[整數]]，其中每行、每列、每個柱及四條{{tsl|en|Space diagonals||空間對角線}}上數字的和均相同，等於立方體的[[幻方常數]]，記為''M''<sub>3</sub>(''n'')。<ref name=":0">{{Cite mathworld|urlname=MagicCube|title=Magic Cube|access-date=2016-12-04|archive-date=2021-03-07|archive-url=https://web.archive.org/web/20210307033550/https://mathworld.wolfram.com/MagicCube.html|dead-url=no}}</ref>若魔術立方體由數列1, 2, ..., ''n''<sup>3</sup>構成，則可以證明其幻方常數為{{OEIS|id=A027441}}
+:<math>M_3(n) = \frac{n(n^3+1)}{2}</math>
+另外，如果每個[[截面 (幾何)|截面]]對角線上的數字之和亦等於幻方常數，則稱此立方體為{{tsl|en|perfect magic cube||完美魔方}}；否則，稱其為{{tsl|en|semiperfect magic cube||半完美魔方}}。數字''n''稱為魔方的階。如果幻方{{tsl|en|broken space diagonal||破碎空間對角線}}上的數字和也等於幻方常數，則稱其為[[泛對角線立方體]]。
+==參見==
-:<math>M_3(n) = \frac{n(n^3+1)}{2}.</math>
-If, in addition, the numbers on every [[截面 (幾何)]] diagonal also sum up to the cube's magic number, the cube is called a {{tsl|en|perfect magic cube||perfect magic cube}}; otherwise, it is called a {{tsl|en|semiperfect magic cube||semiperfect magic cube}}. The number ''n'' is called the order of the magic cube. If the sums of numbers on a magic cube's {{tsl|en|broken space diagonal||broken space diagonal}}s also equal the cube's magic number, the cube is called a [[pandiagonal cube]].
-==Alternate definition==
-In recent years, an alternate definition for the {{tsl|en|perfect magic cube||perfect magic cube}} has gradually come into use. It is based on the fact that a pandiagonal magic square has traditionally been called '''perfect''', because all possible lines sum correctly. This is not the case with the above definition for the cube.
-==[[Multimagic cube]]s==
-{{Update|inaccurate=yes|reason = see the main article more information has been picked up from MathWorld and other sources about the known cubes|date=October 2011}}
-如同[[魔術方塊]]一般， a [[bimagic cube]] has the additional property of remaining a magic cube when all of the entries are squared, a [[trimagic cube]] remains a magic cube under both the operations of squaring the entries and of cubing the entries.<ref name=":0" />  (Only two of these are known, as of 2005.)  A {{tsl|en|tetramagic cube||tetramagic cube}} remains a magic cube when the entries are squared, cubed, or raised to the fourth power.
-==Magic cubes based on Dürer's and Gaudi Magic squares==
-A magic cube can be built with the constraint of a given magic square appearing on one of its faces [http://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-duerer-s-square Magic cube with the magic square of Dürer], and [http://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-gaudi-s-square Magic cube with the magic square of Gaudi]
-==See also==
 {{colbegin||20em}}
+* {{tsl|en|Magic hypercube||魔術超立方體}}
-* [[Perfect magic cube]]
+* {{tsl|en|Magic series||魔術系列}}
-* [[Semiperfect magic cube]]
+* {{tsl|en|Nasik magic hypercube||Nasik魔術超立方體}}
-* [[Multimagic cube]]
-* {{tsl|en|Magic hypercube||Magic hypercube}}
-* {{tsl|en|Magic cube class||Magic cube class}}
-* {{tsl|en|Magic series||Magic series}}
-* [[Asymptotic magic hyper-tesseract]]
-* {{tsl|en|Nasik magic hypercube||Nasik magic hypercube}}
 * {{tsl|en|John R. Hendricks||John R. Hendricks}}
 {{colend}}
-== References ==
+== 注釋 ==
+{{reflist}}
-<references />{{Magic polygons}}
-==External links==
+==外部連結==
 * {{MathWorld |urltitle=MagicCube |title=Magic Cube}}
-* Harvey Heinz, [http://members.shaw.ca/hdhcubes/index.htm All about Magic Cubes]
+* Harvey Heinz, [http://members.shaw.ca/hdhcubes/index.htm All about Magic Cubes] {{Wayback|url=http://members.shaw.ca/hdhcubes/index.htm |date=20181124125113 }}
-* Marian Trenkler, [http://math.ku.sk/~trenkler/aa-cub-01.pdf Magic p-dimensional cubes]
+* Marian Trenkler, [http://math.ku.sk/~trenkler/aa-cub-01.pdf Magic p-dimensional cubes] {{Wayback|url=http://math.ku.sk/~trenkler/aa-cub-01.pdf |date=20160303231904 }}
-* Marian Trenkler, [http://math.ku.sk/~trenkler/05-MagicCube.pdf An algorithm for making magic cubes]
+* Marian Trenkler, [http://math.ku.sk/~trenkler/05-MagicCube.pdf An algorithm for making magic cubes] {{Wayback|url=http://math.ku.sk/~trenkler/05-MagicCube.pdf |date=20160303171656 }}
-* Marian Trenkler, [http://www.imi.ajd.czest.pl/zeszyty/zeszyt13/Trenkler.pdf On additive and multiplicative magic cubes]
+* Marian Trenkler, [https://web.archive.org/web/20120321091253/http://www.imi.ajd.czest.pl/zeszyty/zeszyt13/Trenkler.pdf On additive and multiplicative magic cubes]
-* [http://sites.google.com/site/aliskalligvaen/home-page Ali Skalli's magic squares and magic cubes]
+* [http://sites.google.com/site/aliskalligvaen/home-page Ali Skalli's magic squares and magic cubes] {{Wayback|url=http://sites.google.com/site/aliskalligvaen/home-page |date=20130525015716 }}
-[[Category:Magic squares]]
+[[Category:魔術正方體]]
+[[Category:趣味數學]]
+[[Category:幻方]]