# Three Cube Packing for All Dimensions
## 3 Discussion  
There are three \(d\)-cubes, that is, three variables. In the previous proofs [20; 21; 22; 23], the apparent substitution \(z^{d}=1-x^{d}-y^{d}\) is used to achieve only two variables. As a result of it, the domain boundaries \(x^{d}+y^{d}=1\) and \(x^{d}+2y^{d}=1\) change as the dimension changes. For example, in [23], the shape of the curve \(C:x^{d}y^{d}+y^{2d}-y^{d}+(x^{2}-y^{2})^{d}=0\) is analyzed depending on the dimension, concluding: \"...the shape of the curve \(C\) is similar...\". The word \"similar\" (but not the same) is essential. The curve \(C\) is also dimension dependent. (The curve \(C\) in previous proofs has the same role as our curve \(PB\).) \begin{table}
\begin{tabular}{c c c} \hline \hline \(d\) & **Signs** & **Sign Changes** \\ \hline
2 & \(-+++-\) & 2 \\  & \(++++-\) & 1 \\ \hline
3 & \(-++-\) & 2 \\  & \(+++-\) & 1 \\ \hline
4 & \(-++++-+--\) & 4 \\  & \(++++-++--\) & 3 \\ \hline
5 & \(-++---+-\) & 4 \\  & \(+++--+--\) & 3 \\ \hline
6 & \(-++++-++++----\) & 6 \\  & \(++-++++-++++--\) & 5 \\ \hline
7 & \(-++++-+----\) & 6 \\  & \(+++--+----\) & 5 \\ \hline
8 & \(-++-++++----+--\) & 8 \\  & \(+++-++++-++++----\) & 7 \\ \hline
9 & \(-++++----+-+-\) & 8 \\  & \(+++-+++----+--\) & 7 \\ \hline
10 & \(-++++++++-----+----\) & 10 \\  & \(++-++++-++++-++++-++++-++----\) & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of Sturm chain for \(2\leq d\leq 10\). It is difficult, if not impossible, to perform calculations in all dimensions at once when the curve \(C\) and the boundaries change with the change in dimension. Our substitutions \(y=kx\), \(z=mx\) and \(x^{d}=(k^{d}+m^{d}+1)^{-1}\) lead to the boundaries \(m=k\), \(k=1\), \(m=1-k\), which are simpler than the boundaries in the previous proofs, and do not depend on the dimension (see Figure 2). The curve \(PB\): \(1=k(k+m)\) is also the same for all dimensions, and it is much simpler than the curve \(C\). Such huge simplifications are not free. For dimensions from 2 to 10, the volume function \(x^{d}+\frac{x^{d}+1}{y}\) changed to \(\frac{(k+1)(k+m)}{k^{d}+m^{d}+1}\), but the stable domain more than compensates for this trade-off. The \"dimensionless\" and simplicity of both the boundaries and the \(PB\) curve allowed us to achieve the following:  
* The preceding proofs cover only one dimension at a time and only for some dimensions less than 10. We present the results for all dimensions \((d\geq 2)\) in about the same number of pages. * The previous proofs are based mainly on numerical calculations. There are significantly fewer numerical calculations in our proofs. * The method of the previous proofs would have a precision problem in higher dimensions. Our proofs are not affected by this problem. * The presented results are shorter. The final result of the preceding proofs is a system of two equations with two variables. Our result is the single one-variable equation, and it has the same or lesser degree. For example, the final system of equations for the fifth dimension from [23]: \(7x^{6}y^{5}+12 xy^{10}-6 xy^{5}+5x^{5}y^{6}+10y^{11}-5y^{6}+(x^{2}-y^{2})^{4}(2 x^{3}-10y^{3}-12 xy^{2})=0\), \(x^{5}y^{5}+y^{10}-y^{5}+(x^{2}-y^{2})^{5}=0\). Our final equation: \(15k^{7}-10k^{6}-5k^{5}-4k^{3}+8k^{2}+3k-4=0\). Our final equation is even simpler for dimensions greater than 10. For example, if \(d=17\), then \(32k^{17}+17k^{16}=1\). Although the equations are different, we can confirm that the volumes and the lengths of the edges published in [20; 21; 22; 23] are the same as ours. * Compared to the previous results, we guarantee that the final equation has exactly one solution in the interval. * In [23], it was conjectured that there is only a single maximal packing for each dimension greater than 10, and in these packings, the two smallest \(d\)-cubes are the same. We proved this conjecture. The global maximum of \(G\) occurs on \(PB\) (it means two different maximum packings \(W_{1}\) and \(W_{2}\), even though they use the same \(d\)-cubes) only if \(2\leq d\leq 10\).