A semiperfect magic cube, sometimes also called an "Andrews cube" (Gardner 1976; Gardner 1988, p. 219) is a magic cube for
which the cross section diagonals do not sum to
the magic constant. Some care is needed with terminology,
as some authors drop the "semiperfect" and refer to such cubes simple as
"magic cubes" (e.g., Benson and Jacoby 1981, p. 4).

A semiperfect magic cube of order 3 has magic constant 42. It must be associative with opposite elements summing to (Andrews 1960, p. 65) and have as its center (Gardner 1976; Benson and Jacoby
1981, p. 4; Andrews 1960, p. 65). Hendricks (1972) proved that there are
four distinct semiperfect magic cubes excluding rotations and reflections (Gardner
1976; Benson and Jacoby 1981, pp. 4 and 11-13), illustrated above. These cubes
were described by Andrews (1960, pp. 66-70), although he seems not to have noted
that they represent all distinct possibilities (Gardner 1976; Benson and Jacoby
1981, p. 4). Order three semiperfect magic cubes are also illustrated by Hunter
and Madachy (1975, p. 31) and Ball and Coxeter (1987, p. 218)

The above semiperfect magic cube of order four (Ball and Coxeter 1987, p. 220) has magic constant 130.

Semiperfect cubes of odd order with and doubly even
order can be constructed by extending the methods used for magic
squares. Pandiagonal semiperfect cubes exist for all orders and all odd (Ball and Coxeter 1987).