Equations for perfect magic 5x5x5 cubes with pandiagonal surface squares, JavaScript required                                                                                                                                                               © 2013, H.B. Meyer
001: 006/007/008/009/010 002: 011/012/013/014/015 003: 016/017/018/019/020 004: 001/006/011/016/021 005: 002/007/012/017/022 006: 003/008/013/018/023
007: 004/009/014/019/024 008: 001/007/013/019/025 009: 005/009/013/017/021 010: 001/010/014/018/022 011: 002/006/015/019/023 012: 003/007/011/020/024
013: 002/008/014/020/021 014: 003/009/015/016/022 015: 004/010/011/017/023 016: 026/027/028/029/030 017: 031/032/033/034/035 018: 036/037/038/039/040
019: 041/042/043/044/045 020: 026/031/036/041/046 021: 027/032/037/042/047 022: 028/033/038/043/048 023: 029/034/039/044/049 024: 026/032/038/044/050
025: 030/034/038/042/046 026: 051/052/053/054/055 027: 056/057/058/059/060 028: 061/062/063/064/065 029: 066/067/068/069/070 030: 071/072/073/074/075
031: 051/056/061/066/071 032: 052/057/062/067/072 033: 053/058/063/068/073 034: 054/059/064/069/074 035: 051/057/063/069/075 036: 055/059/063/067/071
037: 076/077/078/079/080 038: 081/082/083/084/085 039: 086/087/088/089/090 040: 091/092/093/094/095 041: 096/097/098/099/100 042: 076/081/086/091/096
043: 077/082/087/092/097 044: 078/083/088/093/098 045: 079/084/089/094/099 046: 076/082/088/094/100 047: 080/084/088/092/096 048: 101/102/103/104/105
049: 106/107/108/109/110 050: 111/112/113/114/115 051: 116/117/118/119/120 052: 101/106/111/116/121 053: 102/107/112/117/122 054: 103/108/113/118/123
055: 104/109/114/119/124 056: 101/107/113/119/125 057: 105/109/113/117/121 058: 101/110/114/118/122 059: 102/106/115/119/123 060: 103/107/111/120/124
061: 102/108/114/120/121 062: 103/109/115/116/122 063: 104/110/111/117/123 064: 002/027/052/077/102 065: 003/028/053/078/103 066: 001/027/053/079/105
067: 005/029/053/077/101 068: 001/030/054/078/102 069: 002/026/055/079/103 070: 003/027/051/080/104 071: 002/028/054/080/101 072: 003/029/055/076/102
073: 006/031/056/081/106 074: 008/033/058/083/108 075: 006/032/058/084/110 076: 010/034/058/082/106 077: 011/036/061/086/111 078: 012/037/062/087/112
079: 014/039/064/089/114 080: 011/037/063/089/115 081: 015/039/063/087/111 082: 016/041/066/091/116 083: 017/042/067/092/117 084: 018/043/068/093/118
085: 019/044/069/094/119 086: 016/042/068/094/120 087: 020/044/068/092/116 088: 021/046/071/096/121 089: 022/047/072/097/122 090: 023/048/073/098/123
091: 024/049/074/099/124 092: 021/047/073/099/125 093: 025/049/073/097/121 094: 021/050/074/098/122 095: 022/046/075/099/123 096: 023/047/071/100/124
097: 022/048/074/100/121 098: 023/049/075/096/122 099: 005/035/065/095/125 100: 005/050/070/090/110 101: 010/030/075/095/115 102: 015/035/055/100/120
103: 010/040/070/100/105 104: 015/045/075/080/110 105: 004/034/064/094/124 106: 024/044/064/084/104 107: 003/033/063/093/123 108: 023/043/063/083/103
109: 002/032/062/092/122 110: 022/042/062/082/102 111: 001/031/061/091/121 112: 021/041/061/081/101 113: 001/046/066/086/106 114: 006/026/071/091/111
115: 011/031/051/096/116 116: 006/036/066/096/101 117: 011/041/071/076/106 118: 001/032/063/094/125 119: 005/034/063/092/121 120: 021/042/063/084/105
121: 025/044/063/082/101                                 
c001
c002
c003
c004
c005
c006
c007
c008
c009
c010
c011
c012
c013
c014
c015
c016
c017
c018
c019
c020
c021
c022
c023
c024
c025
c026
c027
c028
c029
c030
c031
c032
c033
c034
c035
c036
c037
c038
c039
c040
c041
c042
c043
c044
c045
c046
c047
c048
c049
c050
c051
c052
c053
c054
c055
c056
c057
c058
c059
c060
c061
c062
c063
c064
c065
c066
c067
c068
c069
c070
c071
c072
c073
c074
c075
c076
c077
c078
c079
c080
c081
c082
c083
c084
c085
c086
c087
c088
c089
c090
c091
c092
c093
c094
c095
c096
c097
c098
c099
c100
c101
c102
c103
c104
c105
c106
c107
c108
c109
c110
c111
c112
c113
c114
c115
c116
c117
c118
c119
c120
c121
c122
c123
c124
c125
sum
A perfect magic 5x5x5-cube consists of 125 integers, such that any 5 entries of any straight line and of any diagonal (spatial as well as plane) have sum 315. The first such cube with entries 1,2,...,125 was found by W. Trump in November, 2003. If, in addition, all 6 magic surface squares are pandiagonal (the broken diagonals sum up to 315, too), then the cube is a PMCP, a perfect magic cube with pandiagonal surface squares. A 5x5x5-cube is "centralsymmetric", if any two entries in position centralsymmetric to the cube midpoint always have sum 126. - This website contains a proof of following:
Theorem   Every perfect magic 5x5x5 cube with pandiagonal surface squares is "centralsymmetric".
Proof
  The 125 cube entries are called c001, c002,...,c125. Enumeration is done from front side to behind, and in every 5x5-square pane, parallel to the front pane, from top left to right below, as shown by the (blue colored) illustration right above; additionally the cube panes are shown from top (yellow) and from the right side (green). From all "sum=315" equations defining a PMCP, a certain choice of 121 (linearly independent) equations no. 001 til no. 121 was made, each of which is represented by 5 places for entries with sum 315, two buttons (red and green) and a "counting text box". Clicking a red button followed by pressing the space key, the attached equation (*+*+*+*+*=315) will be subtracted one time (and when space key is held down, repeatedly), green buttons cause addition of equations. The counting text box shows, how often an equation was added (resp. subtracted for negative counts). The result of all made subtractions and additions is shown in the "display" below. Due to spatial rotations and plane reflections of the cube, it is sufficient, to prove the "centralsymmetric-property" for only 9 entries, namely c001, c002, c003, c007, c008, c013, c032, c033 und c038.
With the above "calculator" the following equations (among others) can be verified:
25c001+25c125=25x126,  50c002+50c124=50x126,   25c003+25c123=25x126,   25c007+25c119=25x126,   50c008+50c118=50x126,   5c013+5c113=5x126,   75c032+75c094=75x126,   75c033+75c093=75x126   und   15c038+15c088=15x126.
When any of the 9 (grey colored) buttons c001, c002,...,c038 is clicked, the display shows the claimed attached equation and the counting text boxes of the 121 defining equations tell, how often any particular equation was added or subtracted. By clicking red and green buttons together with proper use of the space key, the entries of all counting text boxes can be brought to zero; the display below then shows the result 0=0. Since all made calculations are reversible, it follows, that the 9 equations to be proved, can be derived from the 121 defining equations and the start equation 0=0 by proper additions or subtractions. The theorem is now proved.
Remark   At W. Trump and at: "Equations for "centralsymmetric" perfect magic 5x5x5-cubes" it is proved, that such cubes cannot consist of the integers 1,2,...,125, exactly; this implies:
Conclusion   Perfect magic cubes with pandiagonal surface squares with entries 1,2,...,125, exactly, do not exist.
Nevertheless, there exist PMCPs with pairwise different entries and 315-sums built with integers from greater intervals than [1,125]. The smallest possible such interval (with midpoint 63) is [-129,255].