18
$\begingroup$

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

Improve this question
$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$
    – Dennis Jaheruddin
    Commented Apr 23, 2018 at 13:11
  • $\begingroup$ @Therandomguy Please post answers as answers, not comments. $\endgroup$
    – David Richerby
    Commented Apr 23, 2018 at 16:11
  • $\begingroup$ 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.] $\endgroup$
    – supercat
    Commented Apr 23, 2018 at 16:16

2 Answers 2

Reset to default
23
$\begingroup$

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 $\frac{1}{2}$.

X-shaped cutout of a cube

Comparison of volumes from different methods:

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

Improve this answer
$\endgroup$
7
  • $\begingroup$ Nice one, particularly how you form one face from the corners. Could you also provide the working how you arrived at the final answer? $\endgroup$
    – Phylyp
    Commented Apr 23, 2018 at 9:46
  • 3
    $\begingroup$ The comparison is nice. Any proof that this solution cannot be improved upon? $\endgroup$
    – Shufflepants
    Commented Apr 23, 2018 at 19:12
  • 1
    $\begingroup$ 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.. $\endgroup$
    – Amit Naidu
    Commented Apr 24, 2018 at 5:38
  • 1
    $\begingroup$ @AmitNaidu There is actually a longer line, going diagonally through two opposite faces and connecting on the edge between the other faces. $\endgroup$
    – w l
    Commented Apr 24, 2018 at 7:56
  • 1
    $\begingroup$ @wl But that line, if shifted sligthtly to a parallel plane, appears not 'straight': it turns by 45° when crossing the cube's edge. $\endgroup$
    – CiaPan
    Commented Apr 24, 2018 at 9:20
5
$\begingroup$

This:

enter image description here

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

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

Improve this answer
$\endgroup$
4
  • $\begingroup$ The previous answer has a greater volume though and is way more simple $\endgroup$
    – Gintas K
    Commented Apr 23, 2018 at 11:33
  • 11
    $\begingroup$ @GintasK Sure, but it's still an interesting approach. $\endgroup$
    – maxathousand
    Commented Apr 23, 2018 at 13:15
  • $\begingroup$ @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... $\endgroup$
    – CiaPan
    Commented Apr 24, 2018 at 8:03
  • $\begingroup$ 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). $\endgroup$
    – CiaPan
    Commented Apr 24, 2018 at 8:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.