Semi-magic square of distinct positive cubes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $1\le i,j\le3$, let $a_{ij}$ be positive integers, with all nine $a_{ij}$ pairwise distinct. A $3\times3$ array of cubes is semi-magic when all row sums and all column sums equal the same integer $M$.

Statement recorded in the supplied source.

Do there exist such $a_{ij}$ and $M\in\mathbb N$ with \[ \sum_{j=1}^{3}a_{ij}^{3}=M\quad(i=1,2,3),\qquad \sum_{i=1}^{3}a_{ij}^{3}=M\quad(j=1,2,3)? \] Construct an example or prove there is none. [S2]

Short English statement

Find a three-by-three array of nine distinct positive cubes with every row and every column having the same sum, or prove no such array exists.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9300006 (AMR-092-0006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tim S. Roberts, Unsolved problems in number theory, logic, and cryptography SquareofCubes problem statement. https://unsolvedproblems.org/index_files/SquareofCubes.htm.