Are you up for a challange?

[rahian2k]

[deleted]

 

[stapel]

I'm sorry, but there appears to have been some miscommunication. This is a tutoring forum, not a recreational math or puzzle challenge group.

If you are needing assistance with this exercise, then please post what you have done and specify where you are having difficulty. But if you're just posting for fun, then please find an appropriate outlet and allow the tutors to invest their time and efforts on students needing assistance.

Thank you for your consideration.

Eliz.

 

[soroban]

Hello, rahian2k!

This should have been posted under "Other topics"

I have an intuitive solution . . . but no rigorous proof.

120 unit squares are arbitarily arranged in a 20 x 25 rectangle (both position and orientation is arbitary).
Prove that it is always possible to place a circle of unit diameter inside the rectangle without intersecting any of the squares.

I considered the worst-case scenario.

Place the 120 unit squares evenly spaced in the rectangle.
. . They are 1 unit apart.
. . The top row is flush to the top of the rectangle
. . There is an empty row at the bottom.

      +-------------------------+
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |                         |
      +-------------------------+

The unit-circle can be centered at any "*" without intersecting a square.

      +-------------------------+
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      | @ @ @ @ @ @ @ @ @ @ @ @ |
      |* * * * * * * * * * * * *|
      +-------------------------+

And I see no way of moving or rotating the squares
. . to eliminate all the possibilities.

 

[Denis]

Looks good to me, Soroban; expanding:

to block top empty row, 2nd full row needs to be pushed up a little; .0000000001 will do!

then 3rd full row needs to be pushed up; .0000000002

and so on: no way can you block off bottom