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The A4 European paper format is designed such that if you cut in halve the longest side you would obtain two papers where the ratio of the longest side of each is the same as the original paper.

Is it possible to build an A4 paper that satisfies this objective with a ruler and a pair of scissors?

I don't understand what is meant by ratio of the longest side of each. Can someone also give me a hint on what to do to prove this mathematically?

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  • $\begingroup$ It's about constructing $\sqrt2$. $\endgroup$ – Angina Seng Oct 3 '19 at 19:32
  • $\begingroup$ why is it about constructing $\sqrt {2}$ ? $\endgroup$ – WindBreeze Oct 3 '19 at 19:34
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    $\begingroup$ "Ratio of the longest side" is meaningless, but I'm guessing they meant to write "ratio of the longest side to the shortest side". In other words, the two half sheets are the same shape as the original sheet. $\endgroup$ – bof Oct 3 '19 at 19:34
  • $\begingroup$ Note that $\frac {2x}{\sqrt2x}=\frac{\sqrt2x}x$. $\endgroup$ – Don Thousand Oct 3 '19 at 19:35
  • $\begingroup$ I do not understand what in the context of the problem makes it's about constructing $\sqrt{2}$ ? $\endgroup$ – WindBreeze Oct 3 '19 at 19:37
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As @LordSharktheUnknown notes in his comment, the question is essentially asking you to, given a length $x$, compute a length $\frac x{\sqrt2}$. This is because if the long side of the paper is $x$, and the short side $\frac x{\sqrt2}$, we have the ratio as $\sqrt2$, and the ratio of the cut rectangle as $$\frac{\frac x{\sqrt2}}{\frac x2}=\sqrt2$$So, we can construct this length by drawing length $x$ on the side of a large piece of paper on the corner, and then folding the corner such that the folded paper just hits the end of the drawn length. After unfolding, the folded line should be length $\sqrt2 x$. We can easily halve this length to get the desired length for the short side.

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  • $\begingroup$ I'm having trouble seeing why the shorter side would be $\dfrac{x}{\sqrt{2}}$ and why $\sqrt{2}$ is the ratio? $\endgroup$ – WindBreeze Oct 3 '19 at 19:48
  • $\begingroup$ Try reading the first part of my answer again, $\endgroup$ – Don Thousand Oct 3 '19 at 19:49
  • $\begingroup$ I don't seem to be able to grasp why does having a length of x implies an width $\dfrac{x}{\sqrt{2}}$? Where does the $\sqrt{2}$ comes from? $\endgroup$ – WindBreeze Oct 3 '19 at 19:54
  • $\begingroup$ Did you read the sentence beginning with 'This is because"? $\endgroup$ – Don Thousand Oct 3 '19 at 19:54
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    $\begingroup$ I think the OP needs a derivation, not just a number $\sqrt{2}$ coming out of nowhere. $\endgroup$ – peter.petrov Oct 3 '19 at 19:56
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  • $\begingroup$ I would have suggested this construction, but OP didn't mention having a compass. $\endgroup$ – Don Thousand Oct 3 '19 at 20:14
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    $\begingroup$ @DonThousand: you may replace the compass with origami: just divide the square into four congruent squares, then bisect twice the $90^\circ$ angle in the bottom left. Diagram updated. $\endgroup$ – Jack D'Aurizio Oct 3 '19 at 20:16
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Suppose the longer side is $ax$ and the shorter side is $x$. Cut the sheet into two.
Now if the longer side is $x$ and the shorter one is $ax/2$ (i.e. if $\frac{ax}{2} \lt x$), we need to have:

$\frac{long\ side}{short\ side} = \frac{ax}{x} = \frac{x}{\frac{1}{2}ax}$

Solve this for $a$ and you get $a=\sqrt{2}$

Of course there is a second case here... After cutting it may turn out that $\frac{ax}{2}$ is still the longer side (i.e. it may turn out that $\frac{ax}{2} \ge x$). You can try to form an equation for this case yourself and see that it leads us to nowhere.

So now the problem becomes this... we pick an arbitrary segment and name its length $x$. The task is now to construct a segment with length $ax = \sqrt{2} . x$ If we do that the problem would be solved. That's what other folks meant by "this problem is about constructing $\sqrt{2}$".

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Answer: No, it is not possible.

The size of an A4 paper is roughly 297mm by 210mm. If you cut 297mm in half (297/2), then 297/2=148.5 becomes your short side of the new paper. The 210mm side becomes the long side of the new paper.

If you check the ratio of the first sheet (divide 297 by 210) and the second sheet (210/148.5), you will find out that there is a small difference in the ratio between the two papers (the ratio is very close to the square root of 2). This means that achieving the wanted result with scissors and a ruler requires tremendous precision (more on this later).

A bit of research about international paper sizes leads us to consider the square root of 2: https://www.mathsisfun.com/geometry/paper-sizes.html

To obtain a perfect ratio between the two papers, we must follow a rule of international paper sizes. The rule in question goes as follows: if the short side of your paper is of size x, then the long side must be x*sqrt(2). If we take our initial paper 297mm by 210mm, we find out that 210*sqrt(2) roughly equals 296.98484 rather than 297. We should also note that the ratio ((x*sqrt(2))/x) is always sqrt(2). So, we can see that it would be very difficult to make a perfect A4 paper format by hand due to the precision limitations linked to using a ruler and scissors.

Now, repeating myself a little bit, if we divide our perfect 210*sqrt(2)mm by 210mm paper by 2 (divide the long side by 2), we will find that the proportions between the initial paper and the new paper are identical. The short side of the new paper would be ((210*sqrt(2))/2) and its long side would be of course 210mm. Dividing 210 by ((210*sqrt(2))/2) of course gives us sqrt(2).

Now, our proof:

  1. If A4(long side)=x*sqrt(2) and A4(short side)=x, for x an element of positive rational numbers, then A4(long side) is an element of positive irrational numbers given that a rational number multiplied by an irrational number=irrational number.
  2. Let a set "Ruler"={positive rational numbers from 0 to 30}. Given that A4(long side) is not an element of Ruler, then there is no way to accurately measure A4(long side) with Ruler. Thus, the only "solution" would be to round up or down A4(long side) to render it a positive rational number. But then, the ratio between the two papers would no longer be equal to each other (sqrt(2)), which can be demonstrated as such: A4(long side)/A4(short side)=A4(short side)/(A4(long side)/2)=(2*A4(short side))/A4(long side)-->(A4(long side))^2/(A4(short side))^2=2-->A4(long side)/A4(short side)=sqrt(2). If we assumed that A4(long side) and A4(short side) are positive rational numbers, then there is a contradiction that A4(long side)/A4(short side)=sqrt(2), given that the quotient of two rational numbers cannot be irrational. Therefore, A4(long side) must remain irrational to retain the ratio.
  3. We conclude that A4 paper formats can't be made with a ruler and scissors.
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