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

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.
  • #1
julien
1
0
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
 
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  • #2
It's possible with 7 squares, as shown http://www.stetson.edu/~efriedma/squcosqu/" .
 
Last edited by a moderator:
  • #3
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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