# Three Cube Packing for All Dimensions
## 2 Results
If \(d\geq 7\), then \(2<\frac{\sqrt{2}+1}{2^{1-d/2}+1}=W_{2}(P)\). Therefore, if \(d\geq 7\), then \(W_{2}(P)>W_{AB}\). Assume, for the sake of contradiction, that if \(2\leq d\leq 6\), then \(W_{AB}\geq W_{2}(P)\) at some \(k\in\left[\frac{1}{2},1\right]\). From \(\frac{k+1}{k^{d}+(1-k)^{d}+1}\geq\frac{\sqrt{2}+1}{2^{1-d/2}+1}\) we obtain \(k\geq\frac{\left(\sqrt{2}+1\right)\left(k^{d}+(1-k)^{d}+1\right)}{2^{1-d}+1}-1\). Let \(t(k)\) denote the right-hand side, and \(t(k)\) increases on \(\left[\frac{1}{2},1\right]\). For \(2\leq d\leq 6\), \(t\left(\frac{1}{2}\right)\) is successively approximately equal to \(0.810,0.767,0.810,0.895,0.991\), thus \(k>0.76\). We obtain \(k>0.97\) because \(t(0.76)\) is successively approximately equal to \(0.973,1.054,1.151,1.237,1.303\). The value of \(t(0.97)\) is successively approximately equal to \(1.343,1.704,2.034,2.315,2.540\), which implies \(k>1\), which is a contradiction, since \(k\in\left[\frac{1}{2},1\right]\). Thus, if \(2\leq d\leq 6\) and \(k\in\left[\frac{1}{2},1\right]\), then \(W_{2}(P)>W_{AB}\). \(W_{2}(P)>W_{AB}\) holds for \(d\geq 2\), therefore the global maximum of \(W_{2}\) does not occur on \(AB\). The global maximum of \(W_{2}\) must occur on \(PB\). Proof. : \(AP\) is a part of a line \(m=k\) if \(k\in\left[\frac{1}{2},\frac{\sqrt{2}}{2}\right]\). Substituting it into \(W_{2}\), we obtain \(W_{2}(k,k)=\frac{2k(k+1)}{2k^{d}+1}\). We denote it by \(W_{AP}(k)\), \(k\in\left[\frac{1}{2},\frac{\sqrt{2}}{2}\right]\). \(W_{AP}^{\prime}=\frac{\left(4k+2\right)\left(2k^{d}+1\right)-4d(k+1)k^{d}}{ \left(2k^{d}+1\right)^{2}}\) is never undefined. \(W_{AP}^{\prime}=0\) gives \(2(d-2)k^{d+1}+2(d-1)k^{d}-2k-1=0\). We denote the polynomial on the left-hand side by \(p(k)\). It has one sign change, as the sequence of signs is \(+,+|-,-\) (or \(+|-,-\) if \(d=2\)). Therefore, according to [PERSON] rule of signs, it has exactly one positive real root. If \(d\geq 2\), then \(p(2)=2^{d+1}(3d-5)-5\) is positive. Setting \(2\leq d\leq 5\) in \(p\left(\frac{72}{100}\right)=\frac{1}{2}\left(\left(\frac{18}{2}\right)^{d}(8 6d-122)-61\right)\), we see that it is negative. If \(d\geq 5\), then the exponential term dominates and \(p\left(\frac{72}{100}\right)\) is decreasing. So, if \(d\geq 2\), then \(p\left(\frac{72}{100}\right)\) is negative. Therefore, if \(d\geq 2\), then the only positive root of \(W_{AP}^{\prime}=0\) is greater than \(0.72\). Since \(W_{AP}^{\prime}\left(\frac{1}{2}\right)=\frac{2^{d+1}\left(-3d+2^{d+1}+4\right) }{\left(2^{d}+2\right)^{2}}>0\) for \(d\geq 2\), then \(W_{AP}\) is increasing on \(\left[\frac{1}{2},\frac{\sqrt{2}}{2}\right]\). Therefore, the global maximum of \(W_{AP}\) occurs at point \(P\), which also belongs to \(PB\). It implies, together with Claims 2 and 3, that the global maximum of \(W_{2}\) must occur on \(PB\). The global maximum of \(G\) does not occur at points \(P\), \(B\), \(C\). Proof. : \(W_{1}(B)=W_{1}(1,0)=W_{1}(C)=W_{1}(1,1)=1<1.2<\frac{\sqrt{2}+1}{2^{1-d/2}+1}=W_ {1}(P)\) for \(d\geq 2\). Therefore, the global maximum of \(W_{1}\) does not occur at point \(B\) nor \(C\). The point \((0.76,0.76)\in PC\). If \(d\geq 14\), then the sequence \(W_{1}(0.76,0.76)=\frac{2.076+1}{2.076^{d}+1}\) is increasing. \(G(P)=\frac{\sqrt{2}+1}{2^{1-d/2}+1}<\sqrt{2}+1\geq 2.415\). If \(d=14\), then \(2.416\doteq W_{1}(0.76,0.76)\). Hence, if \(d\geq 14\), then \(G(P)<W_{1}(0.76,0.76)\). The point \((0.72,0.72)\in PC\). If \(11\leq d\leq 13\), then \(G(P)\) is successively less than \(2.313,2.342,2.363\) and \(W_{1}(0.72,0.72)\) is successively greater than \(2.315,2.348,2.373\). Hence, if \(11\geq d\geq 13\), then \(G(P)<W_{1}(0.72,0.72)\). Let \(k_{1}=\frac{\sqrt{2}}{2}+0.0001\) and \(m_{1}=\frac{1}{k_{1}}-k_{1}\). The point \((k_{1},m_{1})\in PB\). If \(2\leq d\leq 10\), then \(W_{1}(k_{1},m_{1})\) is successively greater than  
1.20717, 1.41434, 1.60964, 1.78379, 1.9315, 2.05168, 2.14605, 2.21819, 2.2722011, and \(G(P)\) is successively less than  
1.20711, 1.41422, 1.60948, 1.78362, 1.9314, 2.05155, 2.14597, 2.21816, 2.2722010.