# Can a Unit Square be Covered with Six Squares of Side Length Less than 1/2?

• julien
In summary, the conversation discusses a problem from Newman asking to prove that it is impossible to cover a unit square with 5 squares whose sides have lengths less than 1/2. It is mentioned that this problem can be solved with 7 squares, and a link is provided for a visual representation. The conversation then shifts to discussing the process of proving this statement and asks for any ideas.
julien
Hello,

A problem from Newman, which I posted at
http://www.mymathforum.com/POW/POW1.pdf , asks to show that it is
impossible to cover a unit square with 5 squares whose sides have
lengths <1/2.

It is possible to realize such a covering with 7 squares.

What if we allow us only 6 squares ?

JS,
http://www.mymathforum.com

It's possible with 7 squares, as shown http://www.stetson.edu/~efriedma/squcosqu/" .

Last edited by a moderator:
julien said:
A problem from Newman, which I posted at
http://www.mymathforum.com/POW/POW1.pdf , asks to show that it is impossible to cover a unit square with 5 squares whose sides have
lengths <1/2.
how can we prove that? any ideas?

## 1. How do you cover a unit square?

To cover a unit square, you can use a single square with a side length of 1 unit. This will completely cover the unit square, as the area of the square is equal to the area of the unit square.

## 2. Can you cover a unit square with multiple shapes?

Yes, a unit square can be covered with multiple shapes as long as their combined area is equal to the area of the unit square. For example, you can cover a unit square with four triangles, each with a base length of 0.5 and a height of 1.

## 3. What shapes can be used to cover a unit square?

A unit square can be covered with any shape as long as its area is equal to the area of the unit square. Some common shapes used to cover a unit square include squares, rectangles, triangles, and circles.

## 4. Is it possible to cover a unit square with an infinite number of shapes?

Yes, it is possible to cover a unit square with an infinite number of shapes. For example, you can use an infinite number of circles with a decreasing diameter to cover the unit square. However, the total area of the shapes must still be equal to the area of the unit square.

## 5. What is the minimum number of shapes needed to cover a unit square?

The minimum number of shapes needed to cover a unit square depends on the shape being used. For example, a single square with a side length of 1 unit is the minimum number needed. However, if using circles, the minimum number would be infinite as mentioned in the previous question.

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