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Perfect magic cube
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<h2 id="an-alternative-definition">An alternative definition</h2> <p>In recent years[<i>when?</i>], an alternative definition for the perfect magic cube was proposed by <a href="/facts/John_R._Hendricks/WzLgQcLA">John R. Hendricks</a>. By this definition, a perfect magic cube is one in which all possible lines through each cell sum to the magic constant. The name <a href="/facts/Nasik_magic_hypercube/1qD3jBBv">Nasik magic hypercube</a> is another, unambiguous, name for such a cube. This definition is based on the fact that a <a href="/facts/Pandiagonal_magic_square/700eI57C">pandiagonal magic square</a> has traditionally been called 'perfect', because all possible lines sum correctly.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>This same reasoning may be applied to <a href="/facts/Hypercube/q2mRPbIz">hypercubes</a> of any dimension. Simply stated; in an order <i>m</i> magic hypercube, if all possible lines of <i>m</i> cells sum to the magic constant, the hypercube is perfect. All lower dimension hypercubes contained in this hypercube will then also be perfect. This is not the case with the original definition, which does not require that the planar and diagonal squares be a <a href="/facts/Pandiagonal_magic_cube/f6JlE3pV">pandiagonal magic cube</a>. For example, a magic cube of order 8 has 244 correct lines by the <i>old</i> definition of "perfect", but 832 correct lines by this <i>new</i> definition. </p><p>The smallest perfect magic cube has order 8, and none can exist for double odd orders. </p><p>Gabriel Arnoux constructed an order 17 perfect magic cube in 1887. F.A.P.Barnard published order 8 and order 11 perfect cubes in 1888.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p>By the modern (given by <a href="/facts/John_R._Hendricks/WzLgQcLA">J.R. Hendricks</a>) definition, there are actually six <a href="/facts/Magic_cube_class/p3h7750e">classes of magic cube</a>; <a href="/facts/Simple_magic_cube/bVHENNaH">simple magic cubes</a>, <a href="/facts/Pantriagonal_magic_cube/ngSMpBn8">pantriagonal magic cubes</a>, <a href="/facts/Diagonal_magic_cube/LM2uzh8x">diagonal magic cubes</a>, pantriagonal diagonal magic cubes, <a href="/facts/Pandiagonal_magic_cube/f6JlE3pV">pandiagonal magic cubes</a>, and perfect magic cubes.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h2 id="examples">Examples</h2> <p>1. Order 4 cube by Thomas Krijgsman, 1982; magic constant 130.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <table><tbody><tr><td>Level 1<table><tbody><tr><td>32</td><td>5</td><td>52</td><td>41</td></tr><tr><td>3</td><td>42</td><td>31</td><td>54</td></tr><tr><td>61</td><td>24</td><td>33</td><td>12</td></tr><tr><td>34</td><td>59</td><td>14</td><td>23</td></tr></tbody></table></td><td> </td><td>Level 2<table><tbody><tr><td>10</td><td>35</td><td>22</td><td>63</td></tr><tr><td>37</td><td>64</td><td>9</td><td>20</td></tr><tr><td>27</td><td>2</td><td>55</td><td>46</td></tr><tr><td>56</td><td>29</td><td>44</td><td>1</td></tr></tbody></table></td><td> </td><td>Level 3<table><tbody><tr><td>49</td><td>28</td><td>45</td><td>8</td></tr><tr><td>30</td><td>7</td><td>50</td><td>43</td></tr><tr><td>36</td><td>57</td><td>16</td><td>21</td></tr><tr><td>15</td><td>38</td><td>19</td><td>58</td></tr></tbody></table></td><td> </td><td>Level 4<table><tbody><tr><td>39</td><td>62</td><td>11</td><td>18</td></tr><tr><td>60</td><td>17</td><td>40</td><td>13</td></tr><tr><td>6</td><td>47</td><td>26</td><td>51</td></tr><tr><td>25</td><td>4</td><td>53</td><td>48</td></tr></tbody></table></td></tr></tbody></table> <p> 2. Order 5 cube by <a href="/facts/Walter_Trump/sLEHvykA">Walter Trump</a> and Christian Boyer, 2003-11-13; magic constant 315. </p> <table><tbody><tr><td>Level 1<table><tbody><tr><td>25</td><td>16</td><td>80</td><td>104</td><td>90</td></tr><tr><td>115</td><td>98</td><td>4</td><td>1</td><td>97</td></tr><tr><td>42</td><td>111</td><td>85</td><td>2</td><td>75</td></tr><tr><td>66</td><td>72</td><td>27</td><td>102</td><td>48</td></tr><tr><td>67</td><td>18</td><td>119</td><td>106</td><td>050</td></tr></tbody></table></td><td> </td><td>Level 2<table><tbody><tr><td>91</td><td>77</td><td>71</td><td>6</td><td>70</td></tr><tr><td>52</td><td>64</td><td>117</td><td>69</td><td>13</td></tr><tr><td>30</td><td>118</td><td>21</td><td>123</td><td>23</td></tr><tr><td>26</td><td>39</td><td>92</td><td>44</td><td>114</td></tr><tr><td>116</td><td>17</td><td>14</td><td>73</td><td>95</td></tr></tbody></table></td><td> </td><td>Level 3<table><tbody><tr><td>(47)</td><td>(61)</td><td>45</td><td>(76)</td><td>(86)</td></tr><tr><td>107</td><td>43</td><td>38</td><td>33</td><td>94</td></tr><tr><td>89</td><td>68</td><td>63</td><td>58</td><td>37</td></tr><tr><td>32</td><td>93</td><td>88</td><td>83</td><td>19</td></tr><tr><td>40</td><td>50</td><td>81</td><td>65</td><td>79</td></tr></tbody></table></td><td> </td><td>Level 4<table><tbody><tr><td>31</td><td>53</td><td>112</td><td>109</td><td>10</td></tr><tr><td>12</td><td>82</td><td>34</td><td>87</td><td>100</td></tr><tr><td>103</td><td>3</td><td>105</td><td>8</td><td>96</td></tr><tr><td>113</td><td>57</td><td>9</td><td>62</td><td>74</td></tr><tr><td>56</td><td>120</td><td>55</td><td>49</td><td>35</td></tr></tbody></table></td><td> </td><td>Level 5<table><tbody><tr><td>121</td><td>108</td><td>7</td><td>20</td><td>59</td></tr><tr><td>29</td><td>28</td><td>122</td><td>125</td><td>11</td></tr><tr><td>51</td><td>15</td><td>41</td><td>124</td><td>84</td></tr><tr><td>78</td><td>54</td><td>99</td><td>24</td><td>60</td></tr><tr><td>36</td><td>110</td><td>46</td><td>22</td><td>101</td></tr></tbody></table></td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Magic_cube_classes/p3h7750e">Magic cube classes</a></li> <li><a href="/facts/Nasik_magic_hypercube/1qD3jBBv">Nasik magic hypercube</a></li> <li><a href="/facts/John_R._Hendricks/WzLgQcLA">John R. Hendricks</a></li></ul> <ul><li>Frost, A. H. (1878). "On the General Properties of Nasik Cubes". <i>Quart. J. Math</i>. 15: 93–123.</li> <li>Planck, C., The Theory of Paths Nasik, Printed for private circulation, A.J. Lawrence, Printer, Rugby,(England), 1905</li> <li>H.D, Heinz & J.R. Hendricks, <i>Magic Square Lexicon: Illustrated</i>, hdh, 2000, 0-9687985-0-0</li></ul> <h2 id="external-links">External links</h2> <ul><li>Frankenstein, G. (1878). <a href="http://www.multimagie.com/English/Frankenstein.htm">"A big puzzle"</a>.</li> <li>Trump, Walter. <a href="http://www.trump.de/magic-squares/magic-cubes/cubes-1.html">"Perfect magic cube of order 6 found"</a>.</li> <li><a href="https://web.archive.org/web/20040224223239/http://perso.club-internet.fr/cboyer/multimagie/English/Perfectcubes.htm">Christian Boyer: Perfect magic cubes</a></li> <li><a href="http://mathworld.wolfram.com/news/2003-11-18/magiccube/">MathWorld news: Perfect magic cube of order 5 discovered</a></li> <li><a href="http://members.shaw.ca/hdhcubes/cube_perfect.htm">Harvey Heinz: Perfect Magic Hypercubes</a></li> <li><a href="http://www.magichypercubes.com/Encyclopedia/">Aale de Winkel: The Magic Encyclopedia</a></li> <li><a href="http://members.shaw.ca/hdhcubes/cube_update-1.htm#Pandiagonal%20impossibility%20proof">Impossibility Proof for doubly odd order Pandiagonal and Perfect hypercubes</a></li> <li>Most-perfect cube <a href="https://oeis.org/A270205">https://oeis.org/A270205</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>W., Weisstein, Eric. "Perfect Magic Cube". mathworld.wolfram.com. Retrieved 4 December 2016.{{cite web}}: CS1 maint: multiple names: authors list (link) <a href="http://mathworld.wolfram.com/PerfectMagicCube.html" target="_blank">http://mathworld.wolfram.com/PerfectMagicCube.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Alspach, Brian; Heinrich, Katherine. "Perfect Magic Cubes of Order 4m" (PDF). Retrieved 3 December 2016. <a href="/wiki/Brian_Alspach" target="_blank">/wiki/Brian_Alspach</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weisstein, Eric W. (12 December 2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. ISBN 9781420035223. <a href="9781420035223" target="_blank">9781420035223</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Pickover, Clifford A. (28 November 2011). The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions. Princeton University Press. ISBN 978-1400841516. <a href="978-1400841516" target="_blank">978-1400841516</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Perfect Magic Cubes". www.trump.de. Retrieved 4 December 2016. <a href="http://www.trump.de/magic-squares/magic-cubes/cubes-1.html" target="_blank">http://www.trump.de/magic-squares/magic-cubes/cubes-1.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Magic Cube Timeline". www.magic-squares.net. Retrieved 4 December 2016. <a href="http://www.magic-squares.net/c-t-htm/C_timeline.htm" target="_blank">http://www.magic-squares.net/c-t-htm/C_timeline.htm</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>"Magic Cubes Index Page". www.magic-squares.net. Retrieved 4 December 2016. <a href="http://www.magic-squares.net/magic_cubes_index.htm" target="_blank">http://www.magic-squares.net/magic_cubes_index.htm</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"Magic Cube Timeline". www.magic-squares.net. Retrieved 4 December 2016. <a href="http://www.magic-squares.net/c-t-htm/C_timeline.htm" target="_blank">http://www.magic-squares.net/c-t-htm/C_timeline.htm</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>"Magic Cubes Index Page". www.magic-squares.net. Retrieved 4 December 2016. <a href="http://www.magic-squares.net/magic_cubes_index.htm" target="_blank">http://www.magic-squares.net/magic_cubes_index.htm</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>"Archived copy". Archived from the original on 4 March 2016. Retrieved 28 January 2012.{{cite web}}: CS1 maint: archived copy as title (link) <a href="https://web.archive.org/web/20160304034747/http://www.pythagoras.nu/pyth/nummer.php?id=253" target="_blank">https://web.archive.org/web/20160304034747/http://www.pythagoras.nu/pyth/nummer.php?id=253</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>