Cut the square cloth!

Question

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 $x$?

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

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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!

Answer 1

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

(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 n², 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

Answer 2

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.

Answer 3

Is the answer

1

Reason:

The cloth can be cut into 2 pieces like this

Answer 4

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