Lemma 1: In a magic square of order 4, the sum of the corners is
.
Proof: Add together each edge of the square and the two diagonals. This covers the square entirely, and each corner twice again. This adds to
, so twice the corner sum is
.
Lemma 2: In a magic cube of order 4, the sum of any two corners connected by an edge of the cube is
.
Proof: Call the corners
and
. Let
,
, and
,
be the corners of any two edges of the cube parallel to
. Then
,
, and
are all the corners of magic squares. So
;
;
.
Let
(= 130) be the sum of a row. Consider a corner
. There are three corners connected by an edge to
. Each must have the same value
, contradicting the requirement that all values are different. Thus, there is no magic cube of order 4.
QED. (Proof by Richard Schroeppel, 1972)
