In a letter dated January,
1976 E. G. Straus sent this author a detailed construction of a 7 × 7 × 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
Aijk(2)
= xi+ 2xj− 3xk, Aijk(−2)= xi− 2xj− 3xk,
Aijk(3)
= xi+ 3xj+ 2xk
where (x1,…,x7) = (0,1,…,y) and arithmetic is (mod7).
In this paper we superimpose 6 orthogonal Latin cubes of order 7 where each
ordered triple (000,001,…,666) occurs in every one of the 6 possible positions to form
20 separate Straus cubes.
