# Obtain four equal parts with a single cut

It's easy to obtain four equal parts from a rectangle with three cuts: just make four strips. It's also easy to obtain four equal parts from a rectangle with two cuts: make a horizontal and a vertical cut. Is it possible to take a sheet of paper (not necessarily rectangular) and cut it in four equal parts with just a cut? The sheet cannot be folded, and the cut must be a continuous line.

• I found this problem in an old file of mine, without the answer: or better, in the file I wrote "in the picture below a solution is shown. As you can see, the cut is a straight line"; but there was no picture, and I don't remember the solution I had. Now I managed to solve the problem with a single cut, but this is not a straight line: such an answer is welcome, anyway. – mau Jul 6 '17 at 9:39
• (by the way, I manage to find the original reference. Two possible answers are those by Tom and William-Nathanael, but there is another one more interesting at least in my opinion. I will eventually post it here if nobody finds it. – mau Jul 6 '17 at 10:23
• What's a counts as a "sheet" for this question? Does it have to be continuous? and convex? can a "sheet" be curved without folding? – mr23ceec Jul 6 '17 at 13:57
• it must be continuous, otherwise it is not a single sheet; it needn't be convex. – mau Jul 6 '17 at 17:32

## 6 Answers

Okay, I don't know how people here create images that fast, but I get a more sheet-like solution.

Cut through the x-axis:

Roll the paper into a cylinder such that the two edges that go down the inside and the outside are aligned and that there are 4 complete turns in the cylinder. Cut down the point where the edges align.

That's true provided you have an ideal sheet of paper (area but no thickness). Real-world paper having thickness means the inner turn would be slightly shorter than the next one out... and so on to the outside. With a paper thickness of t the difference in width from the outside to the inside piece would be 6t. In the real world, you could buckle the paper while rolling it such that the inside turn had 6t more than 1 rotation in it.

This would work for more or less than 4 as well, though there would be an upper bound where you couldn't roll the paper and accommodate the extra length for the innermost turn.

With a straight line cut one answer is:

To cut the green snake shape, which is not rectangular though not 'sheet like' as below:

I got this answer but it's not a straight line:

• Well, it says just continuous, so your solution looks fine (it you don't look at comments) – Zizy Archer Jul 6 '17 at 13:19

Well, if the paper doesn't have to be rectangular, you can cut an equilateral triangle into four smaller ones. Go from the midpoint of each side to the next. That's a continuous line, although it turns 120 degrees.

One continuous line, but not straight. Produces 4 equal parts.

• That was my own solution :) – mau Jul 6 '17 at 16:12
• I think it's a little unclear whether a cut which becomes parallel the edge of a piece of material is really "continuous", but if one observes that a cut can produce "four equal pieces" while also producing other pieces (a key observation necessary for your solution) it's possible to change the solution so that cuts always pass through "live" material and only cross other cuts at 90-degree angles. – supercat Jul 6 '17 at 21:09
• IMO the wording of the third sentence of the OP (the actual question) does not support a solution that produces four equal parts and a different sized fifth part... – DJohnM Jul 7 '17 at 5:18
• The cut is not a line! – JDługosz Jul 7 '17 at 7:18