0+9=9 | pair {0,9} |
1+8=9 | pair {1,8} |
2+7=9 | pair {2,7} |
3+6=9 | pair {3,6} |
4+5=9 | pair {4,5} |
0+3=3 | pair {0,3} |
1+1=3 | pair {1,2}. |
where {m} means that m is the 4-adic expression. As well, we define [m] which explicitly means that m is decimal expression. With these expression,
In the above matrix, the summation of the elements of any column, row and diagonal vectors has a unique value {12}. We call this characteristic of a matrices "Magic square condition".
Magic square condition:"The summation of the elements of any column, row and diagonal vectors has a unique value." |
We prepare a dozen of such matrices and name them by A0,A1,A2,B0,B1,B2,C0,C1,C2,D0,D1,D2. With the dozen matrices, we make four matrices A,B,C,D whoes elements are four-figure 4-adic numbers as follows:
For these combined matrices, the "magic square condition" (for An, Bn, Cn, Dn) is conserved (satisfied by A,B,C,D). If we choose the dozen 4-adic matrices An,Bn,Cn,Dn (n=0,1,2) very carefully, we can obtain a magic cube. The following is the concrete example. All elements are expressed by 4-adic numbers. For example, you can choose the matrix which has only numbers in a place of {100} in 4-adic. In the columns, row, diagonal vectors of the choosen matrix, one of (0,1,2,3), (0,0,3,3), (1,1,2,2) appears.
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