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I was mucking around in the storeroom today when I should have been working. I had a square $20\times20$ cm piece of cardboard, and a roll of masking tape I'd been using (I can't remember the tape's width, but it was an integer number of centimetres). I started mindlessly wrapping the tape around the cardboard...

When I began, I put the tape (sticky side down) with its cut edge entirely on one side of the square and began wrapping. If I hit an edge or a corner, I folded the tape over so as to leave no tape hanging over the edges of square, and none of the sticky side exposed, and continued wrapping. When I finished and cut the tape, the new cut edge was again entirely on one side of the square. On starting and finishing, I cut the strip of tape perpendicular to its long axis.

When my boss entered the room I threw the cardboard into the compactor, but not fast enough, as he caught a glimpse and asked how much tape I had wasted. I didn't know, so he said that unless I can tell him exactly how much tape I used, he will fire me!

Before he walked in, I did take a photo (see below) of one side of the cardboard, but I'm lost - the cardboard and tape were exactly the same colour...

Is it possible to figure out the exact length of tape I used?


Diagram


This is a purely geometrical puzzle, with a unique solution. Do not assume lengths and angles in the image are to scale. The cardboard is an opaque infinitesimally thin perfect square with two sides (front and back), the tape is an opaque infinitesimally thin foldable but otherwise inflexible rectangle with one sticky side, and the lines within the square represent borders either between tape and tape or tape and cardboard. If lines appear to intersect, they do. If you're confused, it might be worth playing with some real tape to see how the edges and corners work.

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    $\begingroup$ It's been a long time since I posted anything what with work commitments and all. So I've come back gently with an easier geometry problem (that I think nevertheless qualifies as a puzzle) as a bit of a change from my previous big multistep puzzles, most of which were never solved. I hope I'm not too rusty and it turns out to be an OK puzzle... $\endgroup$
    – Anon
    Sep 18 at 15:55
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    $\begingroup$ Partial answer: here is my reconstruction of the path of the tape; I don't see how to uniquely determine the tape's length or area though. $\endgroup$ Sep 18 at 18:44
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    $\begingroup$ Wait, can you fold it around the corner like that? I suppose that has to be the case. My solution was overly constrained. Mmm. $\endgroup$
    – Dr Xorile
    Sep 18 at 19:18
  • $\begingroup$ @2012rcampion Nice picture! $\endgroup$
    – Anon
    Sep 18 at 21:48
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Please tell your boss that you used (I think):

$400/\sqrt{21}\approx 87.287$cm of tape that was 8cm wide

@2012rcampion did a better diagram of the wrapping than mine, so I'm including that here:

Unwrapped wrapping

Then I built a little spreadsheet with the various geometrical constraints:

Geometry

Referring to this diagram to do the calculations gives us the following:

$w' = w/\cos(\theta)$ where $w$ is the width of the tape. So $x = 20-2w'$. Then $y = 20\frac{w'-x}{w'}$ and $z = 20\frac{x}{w'}$ (because of similar triangles).
This immediately constrains $w$ to either 7 or 8.
Next I work out $k$ and compare it with $x$. Since I think $k>x$ this leaves $w$ as 8. Woohoo! Now we just need some geometry.
Going back to @2021rcampion's diagram, you'll see that pythag will get us there. The height is 80. The width is $20 + k + w'$.
$w' = 40/\sqrt{21}$.
$k$ is slightly more interesting. But with a bit of fiddling about with trig, it comes to $120\sqrt{21}-20$. This all cancels in a pleasant way, which makes me think there's an easier way to see this. But it all leads to the answer given.


Edit with logic for why the linking is as it is:

1. The tape must be (at least) $w$ wide, where $w=w'\cos(\theta)$ because of the wide stripe down the middle. 2. Also $\theta<45^{\circ}$ or we wouldn't be able to get that stripe to hit the bottom with some space left over (quite a bit less, obvs, but $45^{\circ}$ is enough to get us going).
3. That means that the little bit in the bottom right must be the start and end of the tape with cardboard between them, since the tape has to be running parallel to the broad stripe in the middle.
4. And it means that the lower of those must the be start because it must be partially covered by the broad stripe in the middle (recall that the ends are fully on the front surface, so the longer one must be the full width (since it's the last to be stuck down it's on top), and the shorter one must be the first and partially covered)).
5. Now work backwards from the end of the tape. Plainly, the last piece wraps around and connects with the broad stripe in the middle. Wrap around again, and there has to be a space between the broad stripe down the middle and the next piece. Hence the middle stripe I've labelled "Cardboard" must, in fact, be cardboard. (Another way to see this: Note that if the cardboard extended to the right, that final piece would wrap around leaving a gap. So a similar gap must exist to the left of the stripe in the middle.)
6. Finally, consider the dotted blue parallel lines in my diagram. The one $k$ away from the left comes from the right hand side of the exposed start of the tape. The one $x$ away comes from the right hand side of the broad stripe down the middle. That means that $k>x$.

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    $\begingroup$ Very very nice! The corner folding was confusing even making the puzzle, and had me playing with actual tape. There are indeed a few ways to see the solution. One that might be easier is once you trg j naq gur urvtug 80, gurer'f n eryngviryl angheny jnl bs rkcerffvat gurgn va grezf bs j, juvpu zrnaf gevt jvyy uryc lbh pnyphyngr gur ulcbgrahfr zber rnfvyl. Ohg vg'f nyy fjvatf naq ebhaqnobhgf... V qvqa'g guvax bs gung zber rssvpvrag jnl bs svaqvat j. $\endgroup$
    – Anon
    Sep 18 at 21:57
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    $\begingroup$ One nitpick: I think your answer does technically assume the image is all to scale, so a brief explanation why everything has to link up as it does in @2012rcampion's picture would be the only thing still technically missing in the answer. Nevertheless, it is in substance correct so I've given the checkmark. $\endgroup$
    – Anon
    Sep 18 at 22:06
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    $\begingroup$ Awesome explanation! I hope the puzzle wasn't too boring to solve... $\endgroup$
    – Anon
    Sep 20 at 21:18
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    $\begingroup$ I thought it was interesting because intuitively it didn't feel (to me at least) that it was constrained enough to have a unique solution $\endgroup$
    – Anon
    Sep 20 at 21:26
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    $\begingroup$ Yeah, great puzzle. Your long hiatus was worth the wait! It's a nice mixture of visual intuition etc that's needed to get it out. $\endgroup$
    – Dr Xorile
    Sep 20 at 21:37

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