# Cut the square cloth!

I have a square cloth with side length $$x$$ cm, and I am going to cut it into at least $$n$$ squares with side length $$1$$ cm for my customer, and also you cannot cut the cloth to thinner pieces (reminded by @risky mysteries). I cannot glue any bits of cloth together. What is the minimum value of $$x$$?

Problem inspired from a math test problem in my school (about a month ago).

Thanks to @Jaap Scherphuis, I now know this is an unsolved problem. So of course I still haven't solved it. You can use a computer!

• Is folding valid? – risky mysteries Jul 28 '20 at 4:04
• @riskymysteries Yes, but you can't say a $1$ by $1.1$ rectangle is a $1$ by $1$ square with folding. You need to cut it. – Culver Kwan Jul 28 '20 at 4:09
• Just to make this clear, so is this $x$ should be defined in term of $n$ a.k.a $x = f(n)$? – athin Jul 28 '20 at 5:07
• Isn't this an open problem in mathematics? en.wikipedia.org/wiki/Square_packing_in_a_square – Jaap Scherphuis Jul 28 '20 at 5:51
• This isn't clear: the problem is inspired from a math test problem, but has not been solved by the OP. So there is no definitive answer? This should be closed IMO. – Earlien Jul 28 '20 at 11:14

$$\lceil\sqrt{n}\;\rceil$$

(the ceiling of the square root of n)

Because

If you need n squares which are 1cm wide, you essentially require a piece of cloth with an area of , i.e. a square of length √n.
But given you can't glue pieces together we take the ceiling of √n to ensure we have enough pieces whose length are 1cm

Clearly, $$x \geq \sqrt n$$, otherwise your original square would have less area than the n smaller squares.

In most cases, you will need $$x = \lceil \sqrt n \rceil$$, as that will be the smallest square that allows you to cut the necessary pieces.

However, for $$n=5$$,

You can make do with $$x = 2\sqrt2$$:

This wastes 3 units, which is less than would be wasted for a size 3 square.

This does not continue to work for higher $$n$$. For example, for $$x=3\sqrt2$$, you could fit 13 squares in a similar pattern, but the slightly smaller $$x = 4$$ would have room for 16.

So, in conclusion,

you need $$x = \lceil \sqrt n \rceil$$, except for $$n=5$$, where $$x=2\sqrt 2$$ will work.

1

Reason:

The cloth can be cut into 2 pieces like this

• No, you cannot cut it to thinner pieces. I will add that. – Culver Kwan Jul 28 '20 at 3:29
• @CulverKwan Make sure to acknowledge the edit! – risky mysteries Jul 28 '20 at 3:30
• Ok, I have acknowledge it! – Culver Kwan Jul 28 '20 at 3:35

In both the square, area will be the same.

suppose big square has area, A = x*x

all small square total area will be A = n * 1 *1

so, x*x = n * 1 * 1

x = √n, for all x>=n

• How can you cut for example 2 whole squares from a piece of cloth of size $\sqrt{2}\times\sqrt{2}$? – Jaap Scherphuis Jul 28 '20 at 6:52
• i think solution will work for all x>=n – Amit Huda Jul 29 '20 at 8:41
• You already said that $x=\sqrt{n}$, so how can it be that $x\ge n$? – Jaap Scherphuis Jul 29 '20 at 8:58