Folding paper into corners

Question

Here is a piece of paper:

Fold it once, and you can get a shape with 9 corners:

Starting with a rectangular sheet of paper and folding twice (along any line), what is the largest possible number of corners that may result?

Rule for counting corners: both concave and convex corners count (as you can see in the example), but they must not be "covered" by the rest of the paper. In other words, you are counting the corners of the silhouette.

Credit: Alex C Weiner

Nice puzzle - you may have to ask to fold it thrice. So far 0, 1, 2 folds give 4, 9, 16 corners - looks interesting! — Tom, Jun 28 '16 at 10:26
@Tom Yes please, what is the answer for 3 folds? I have no idea. — Owen, Jun 28 '16 at 19:20
May need to say if can just fold the rectangle or along any line. — Tom, Jun 28 '16 at 20:30
If you make it even thinner, you can fold the 16-solution like this https://nup.pw/QhvFL5.png, giving 28 corners. So, (x+2)^2 is not optimal. — Carl Löndahl, Jun 28 '16 at 21:18
Carl you have missed two concave corners in your diagram, so the answer for three folds must be at least 30. — Penguino, Jun 29 '16 at 04:31
@ev3commander I think the top 2 folds are supposed to be collinear (making it one fold). — Owen, Jun 29 '16 at 22:50
@CarlLöndahl: The 5-pointed star made from a thin strip folded 4 times has: 5 inner triangles, a central pentagon, 5 exterior angles, 2x4 "fold" corners, plus the 9 angles from crossing the two strip ends, for a total of 15 + 5 + 5 +8 +9 = 42, so the "rule" does not hold. — Benjamin, Jun 30 '16 at 00:14
Also, the shape that looks like a star trek badge, but with the two ends extending up and out (3 folds) has 29 angles. — Benjamin, Jun 30 '16 at 01:19
@Owen, right. It was supposed to look collinear :-) Drawing is hard. — Carl Löndahl, Jun 30 '16 at 04:20
@Benjamin Shouldn't it be 25 for three folds? Where does 29 come from? — Eric, Nov 04 '21 at 07:19

Jonathan Allan · Answer 1 · 2016-06-29T03:57:56.400

53

I managed to make

$16$ corners with a thin strip, or $15$ with any rectangle including a square.

Like so:

For a strip:

For a square:

- fold along the red lines
(the black lines on the back are the same as the front except rotated to have the 16th lines in the bottom quarter rather than the top one.)

For a rectangle with less width than height do the same but move the first red line's ends nearer the centre and the second red line's ends nearer the quarter marks (at some point you can make $16$ corners instead though).

edited Jun 29 '16 at 03:57

answered Jun 28 '16 at 09:36

Jonathan Allan

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Is this always possible for an arbitrary n x m rectangle? I feel like it isn't but I wouldn't know how to go about proving that. – RowlandB Jun 28 '16 at 13:40
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@RowlandB It needs to be pretty thin to allow both the hole and the two ends to cross (try it with an A4 sheet!); I have not calculated the limits. – Jonathan Allan Jun 28 '16 at 14:08
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@RowlandB - I added a case for one less corner that works for a square. – Jonathan Allan Jun 28 '16 at 17:38
The strip thing didn't convince me (as not following the spirit of this challenge), but the second part deserves a +1. – pajonk Jun 29 '16 at 08:41
As far as the strip goes, I believe this is possible with any strip whose long to short side ration is greater than exactly 10/sqrt(3). Will post diagram later. – scholtes Jun 29 '16 at 14:21
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https://i.imgur.com/ph8DPCb.png <-- diagram. Showing the limits as the hole shrinks to a point and the sticking out corners reduce to touching an edge. I am assuming that this is minimal when using symmetric equilateral triangles, but you could do some calculus on varying the angles to find out. +1 to JA for considering identifying a case outside the general case. – scholtes Jun 29 '16 at 22:54

score 17 · Answer 2 · answered Jun 28 '16 at 05:53

17

I have managed to come up with

14 corners with two folds.

Following are the images of the paper with each fold.

Unfolded

One fold

Two folds

Corners highlighted

answered Jun 28 '16 at 05:53

CodeNewbie

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score 13 · Accepted Answer · answered Feb 26 '18 at 20:10

13

Using a long and thin rectangular strip, you can get up to

$17$ corners.

Here's how:

answered Feb 26 '18 at 20:10

Afrodeity

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The last three corners come from the triangular hole inside the shape. – Afrodeity Feb 26 '18 at 20:21
1

Looks like a new winner! – Owen Feb 26 '18 at 21:35

score 7 · Answer 4 · answered Jun 28 '16 at 05:50

7

14

by

take a strip of paper (still technically a rectangle) and fold it so the ends cross leaving a triangular hole.

answered Jun 28 '16 at 05:50

Jasen

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Folding paper into corners

4 Answers4