# Squaring the new year

Happy new year!

You are asked to cut a rectangular strip 2015 times longer than its width into pieces that can be reassembled into a square with area equal to that of the original strip.

How many pieces do you need to cut at a minimum?

Hint:

You need no more than $n+1$ pieces to reassemble a rectangle with aspect ratio (long edge length divided by short edge length) smaller than $n^2$ into a square. And $2015$ is smaller than $45^2$...

• Clarification needed: does straight-line-cut mean only straight lines from paper edge to paper edge are allowed, or that each direction-change counts as a single cut? (I.e. If I zickzack through the stripe, but only in the middle of it, is this valid(but counts multiple) or not?) – BmyGuest Jan 1 '15 at 8:41
• @BmyGuest - The description in terms of number of cuts is distracting. I simplified the problem statement. Thanks. – Johannes Jan 1 '15 at 11:26

I can do it in 46 pieces.

Fill the square with horizontal stripes from the top until you have less than 2 units left in height. Then cut a diagonal along the width and cut the remaining piece in 2 as shown in the picture. Left is the 1x2015 cut in 46 pieces. To better show the method, the right picture is a similar solution for a 1x20 stripe cut in 6 pieces.

• 47? I come down to 65, but I might have made a mistake. See above. Also, I wonder: If any mis-tilt works, why not 0? Or does it require a specific tilt to be really exact, which would be what my solution above yields anyway... – BmyGuest Jan 1 '15 at 22:10
• ...ah, not any tilt, but the one given by exactly a strip's width over a full square's length, correct? That is a very specific tilt then. Nice solution. +1 from me (Maybe edit the maths in to have it complete?) – BmyGuest Jan 1 '15 at 22:19
• @FlorianF - nice, +1. However, it is possible to form a square using 46 pieces only... – Johannes Jan 2 '15 at 12:03
• I think I got it. Need to draw the picture. – Florian F Jan 4 '15 at 19:44
• I don't understand your answer. You say to use strips until "less than 2 units left in height", which is 43 strips. Then you're adding 6 pieces which makes 49 pieces in total, not 46. – DisgruntledGoat Jan 5 '15 at 2:14

I might not have the optimum yet, but I think I can at least do a lot better than 4.4 e11...

I can do it with at maximum

96 pieces (--> 65)

# Proof:

As a starting point:

There is a very nice way to convert a rectangle into a square by dissection with only

two cuts. (or 3 pieces)

If you do the following:

I wish I could claim having found that solution myself, but it is actually from

hosted at
New Mexico State University; Department of Mathematical Sciences

Now, this solution has a problem:

It is only valid for rectangles where the two sides fulfil: a < b < 2a
(So a is the shorter side.)

Our "starting point" is a rectangle of size a x 2015a.

But we can make it to fulfil the condition! All we need is to cut stripes until b<2a.

Now we have

a striped rectangle which matches the condition and used up 32 pieces(31 cuts). By applying the cutting in the first image, we will at maximum create 3 pieces hence we will need no more than 3 x 32 = 96 pieces to accomplish the task.

or visually:

Note that X is not splitting the longer side of the rectangle exactly at 1/2, because the longer side is not exactly 63 but 62.96875. The solution is still valid by virtue of the general statement at the beginning of the proof.

Looking at my own answer above, I think there is a very straight-forward way to optimize the solution!

# Improved solution

Starting with the solution above:

It can be seen that

The tiles of the upper half (except the very tip) can of course be merged! This reduces 62 tiles to 31 brings the solution down to 96-31 = 65 tiles.