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This is barely of research interest, but I'd classify it as a curiosity with connections to combinatorics.

The problem is to place integers in an $n \times n \times n$ array so that all $3n^2$ line sums are consecutive integers.

I only have ad-hoc solutions for $n = 2,3,5,7$, and a product construction. For instance, $n = 2$ admits the solution $\left[{\begin{array}{cc} (0,4) & (0,6) \\ (1,4) & (2,5) \\ \end{array}}\right]$, and solutions for $n_1,n_2$ can be easily 'shifted' and 'composed' (sort of like a Kronecker product).

Does anyone notice a construction general enough to work, say, for all primes? Related ideas are welcome too.

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  • $\begingroup$ Remark: The two-dimensional version is easy. In this case, $n$ must be even (or $n=1$). So simply let $A$ be the first layer of the above three-dim solution for $n=2$, which has line sums 0,1,2,3; then for general $n \times n$, form a block diagonal matrix using $A$, $A+2J$, $A+4J$, $\dots$, $A+(n-2)J$. $\endgroup$
    – Peter Dukes
    Commented Apr 13, 2014 at 21:00

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