# How to pack 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with some kind of symmetry?

Recently I stumbled on the problem quoted here about a geometric proof of the AM-GM inequality $$(a_1+\cdots+a_n)^n\ge n^n a_1\cdots a_n$$ by packing $n^n$ rectangular $n$-dimensional boxes of sides $a_1,\dots,a_n$ into a hypercube of side $a_1+\dots+a_n$.

For $n=2$ this is almost trivial $-$ just leave a square of side $|a_2-a_1|$ in the middle and pack four $a_1\times a_2$ rectangles around it. But already for $n=3$ it is highly non trivial, and the quoted article shows in several ways graphically a packing of 27 $a\times b\times c$ blocks into a cube of side $a+b+c$. Presumably this packing was found by a program using trial and error, and it has hardly any symmetries. There should be many non isomorphic packings like this, and I am wondering:

Is there a packing of 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with a higher degree of symmetry? E.g.

• opposite sides of the cube have the same pattern, or

• a certain $S_3$ isomorphism like $(a,b,c)\mapsto (c,b,a)$ results in some rotation of the packed cube? Or if not,

• is there at least a packing where in each direction (or to start with: in just one direction), each one of the layers has three separated holes?
• or, why not, each layer has just two rectangular holes?

As far as the $S_3$ isomorphisms, I'd guess that since the order of the three sides is crucial, if we require wlog $a<b<c$, the isomorphism $(a,b,c)\mapsto (c,b,a)$ should be, if at all, the only one where it might be possible ... Note that because of the special position of the middle length $b$, it also seems hopeless to find a kind of symmetry where $a,b,c$ correspond to the three directions in $\mathbb R^3$.

The idea behind all this is of course that if some high symmetry patterns exist, there might be a tiny chance to generalize them to higher dimensions. Well, tiny... even from $\mathbb R^2$ to $\mathbb R^3$ that is not the case.

• Disclaimer: even though it's holiday time, the idea did not occur to me during packing my suitcase. :) – Wolfgang Aug 9 '18 at 19:14
• You might like to read "Packing Problems and Inequalities" by D.G. Hoffman, in The Mathematical Gardner, books.google.com/… . There are photos. There is also an assertion that there are 21 solutions. – Zach Teitler Aug 9 '18 at 19:41
• There's a nice wooden set available for sale, see this video advertisement: youtube.com/watch?v=lXPfb1uKpnY – Zach Teitler Aug 9 '18 at 19:42
• @ZachTeitler Thank you for that! Now of course it would be interesting to know more about those 21 solutions, but I guess nothing has been published? – Wolfgang Aug 9 '18 at 20:05
• I don't know. The Hoffman article isn't in MathSciNet so I'm not sure how to look for any articles that might refer to it. Something something google :-) – Zach Teitler Aug 9 '18 at 21:02