An extension of E. G. Straus' perfect Latin 3-cube of order 7 (Q1073801)

E. G. Straus, in a private communication to the author in January 1976, gave the construction of a \(7\times 7\times 7\) perfect magic cube written to base 7 with digits 000 to 666. In this construction he superimposed 3 Latin cubes of order 7 (a Latin 3-cube of order 7) to get what may be the lowest possible order of a perfect cube. Before Straus' construction the smallest known perfect cube was of order 8. The Straus cube is perfect in the following way: the sum (2331) of the elements in each minor diagonal and in each pan-diagonal is equal to the sum of the elements of a row in each of the 2 directions in each of the respective squares (layers) that make up this perfect Latin 3-cube of order 7. The sum (2331) of the elements of a row in each direction of the cube is equal to the sum of the elements in each of the 4 major diagonals and the sum on all the diagonals of the cube is the same (namely 2331). The construction of the cube is based on the 3 orthogonal cubes \(A^{(2)}_{ij^ k}=x_ i+2x_ j-3x_ k\), \(A^{(-2)}_{ij^ k}=x_ i-2x_ j-3x_ k\), \(A^{(3)}_{ij^ k}=x_ i+3x_ j+2x_ k\) where \((x_ 1,...,x_ 7)=(0,1,...,y)\) and arithmetic is (mod 7). In this paper 6 orthogonal Latin cubes of order 7 where each ordered triple (000,001,...,666) occurs in every one of the 6 possible positions are superimposed to form 20 separate Straus cubes.