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].
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