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If in every square on an mxn grid there is a number either 0 or 1, and the sum of the numbers in row $i$ is $y_i$, and those of column $j$ is $x_j$. Given $x_i$ and $y_i$ is there a way to reproduce the values of each square (if the data is consistent), and is it unique?

Is it possible with an mxnxo cubic grid knowing the sum of values in every plane (orthogonal to one of the axes)?

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It's certainly not unique, as the examples $\matrix{1&0\cr0&1\cr}$ and $\matrix{0&1\cr1&0\cr}$ show.