Fermat's Magic Cube

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

In 1640, Pierre de Fermat sent a letter to Marin Mersenne about an order-4 magic cube with 64 magic sums out of 76 possible. In a perfect magic cube the rows, columns, pillars, space diagonals, and the diagonals of each n×n orthogonal slice sum to the same number. For the order-4 case, numbers 1 to 64 are required. In 1972, Richard Schroeppel proved that a perfect order-4 magic cube was impossible (see Details).


In 2004, Walter Trump found an order-4 cube with eight more lines that summed to 130.


Contributed by: Ed Pegg Jr (March 2011)
Open content licensed under CC BY-NC-SA



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)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.