# Maximal-volume cube net from unit square paper

Given a square piece of paper (say 1x1), what is the largest cube net you can cut out of it?

('largest' meaning with maximal volume; 'net' meaning it's possible to fold into a cube (somehow))

This puzzle was stolen from 'The bedside book of geometry' (Mike Askew and Sheila Ebbutt) which I happened to find lying around in the house

• First thing that came to mind was actually to make a cube with just a ribs. Too bad that would mean the answer is trivial. Apr 23 '18 at 13:11
• Is there a requirement that there not be any overlapping of excess material? If not, I think the answer would be arbitrarily close to a cube whose sides have a total area equal to that of a square [absent such a requirement, an arbitrarily slender rectangle can be cut and folded to yield a connecting member of arbitrary length, so one could use just about any dissection which converts a square into a cube net, but shrink the pieces just enough to add connecting members.] Apr 23 '18 at 16:16

The largest cube net I can cut out of a 1x1 square paper has a volume of

$\frac{\sqrt{2}}{32} \approx 0.044$

Using the following cutout:

The black area is exactly $\frac{6}{8}$ of the total area (The white triangles are half the area of one face), which means each face has an area of $\frac{1}{8}$, giving a volume of $(\frac{1}{\sqrt{8}})^3 = \frac{\sqrt{2}}{32}$. Alternatively we can notice that the length of the diagonal of a face of the cube is $\frac{1}{2}$.

Comparison of volumes from different methods:

$\begin{array}{l|r|r|r} \text{Method} & \text{Volume} & \text{Approx.} & \text{Percentage} & \text{Wasted paper} \\ \hline \text{Theoretical maximum} & \frac{\sqrt{6}}{36} & 0.068 & 100.0 & 0 \\ \text{wl} & \frac{\sqrt{2}}{32} & 0.044 & 65.0 & \frac{1}{4} \\ \text{CiaPan} & \frac{\sqrt{10}}{100} & 0.032 & 46.5 & \frac{2}{5} \\ \text{Default net along diagonal} & \frac{2\sqrt{2}}{125} & 0.023 & 33.3 & \frac{13}{25} \\ \text{Default net along side} & \frac{1}{64} & 0.016 & 23.0 & \frac{5}{8} \\ \end{array}$

• Nice one, particularly how you form one face from the corners. Could you also provide the working how you arrived at the final answer? Apr 23 '18 at 9:46
• @Phylyp You mean the volume or how I came up with this cutout?
– w l
Apr 23 '18 at 11:28
• The comparison is nice. Any proof that this solution cannot be improved upon? Apr 23 '18 at 19:12
• I think it can be proven from the fact that the longest straight line available is the diagonal of the square paper, and the longest straight line you can draw on a cube is its circumference passing through 4 faces. This cutout maps the former to the latter.. Apr 24 '18 at 5:38
• @wl But that line, if shifted sligthtly to a parallel plane, appears not 'straight': it turns by 45° when crossing the cube's edge. Apr 24 '18 at 9:20

This:

fits a net of a cube with an edge length $a$ into a square with a side of

$$a\,\sqrt{10}$$

so for a square of side length equal $1$ we get a cube with edge length

$$a=\frac 1{\sqrt{10}} \approx 0.316$$

and volume

$$V=a^3 = \frac 1{10\sqrt{10}} \approx 0.0316$$

• The previous answer has a greater volume though and is way more simple Apr 23 '18 at 11:33
• @GintasK Sure, but it's still an interesting approach. Apr 23 '18 at 13:15
• @GintasK Yes, I know. But when I started solving, the previous answer has not been posted yet. Why not post what I've done, just because someone else was faster? Anyway, I hope this site doesn't have any rule against publishing answers which are worse than those already posted... Apr 24 '18 at 8:03
• This question reminded me a similar problem, which I met years ago in some maths newsgroup of Usenet: fit an unfolded net of a cube into an ISO 216 sheet, that is into a 1:sqrt(2) rectangle. The solution above is somewhat similar to that old answer I gave there (which possibly also was not optimal). Apr 24 '18 at 8:04