Square ceiling

anna_gg · 2016-01-23 01:26:41

On a square ceiling 3.5 x 3.5 meters we want to place spot lights with minimum 1 m distance from each other. What is the maximum number of lights we can place?

Relentless · 2016-01-23 01:56:40

Hi

This is a packing problem with a twist. If the spotlights had to be at least 1m from the sides of the ceiling as well, it would be like the common problem of how many unit circles can fit in a square (in this case, the answer is two).

In this problem, however, only the centre of the non-overlapping circles must be inside the square.

Interesting

Relentless · 2016-01-24 15:58:54

Hi,

I have been experimenting with this problem. I have got a maximum of 16 (12 around the sides and 4 in the middle). I am confident that the answer is 16 or 17, and that there are multiple ways to arrange them.
For a distance of 2m the answer is 5

The problem is equivalent to the following:

What is the maximum number of non-overlapping circles of radius 1/2 with a centre inside a square of side length 3.5?

Interestingly, perhaps, I roughly calculated that the 16 circles cover only about 60% of the square's area.

Last edited by Relentless (2016-01-24 18:20:54)

Relentless · 2016-01-25 04:07:34

Hi,

I found a way to fit 17 lights. There is enough space available for 18, but I would be astonished if it was usable.
I'm not really sure how to describe the (my) solution without an image. Basically:
There is a light at each corner.
Two of the corner lights opposite each other have 1.5m gaps on either side, and then there are three lights in a row 1m apart on each side (including the other corner).
There is a light as close as possible to both corner lights with a gap on either side. It looks like these lights are sqrt(2)/2 metres up/down and sqrt(2)/2 metres across from the corner lights, but that is just a guess based on physical measurement.
There is a light as far across as possible from the two corner section lights, i.e. 1m across and 1m up from one of the other corners.
Finally, there is room for two more lights as close as possible to each corner section light (those lights have a calculable distance that is about 2.95m)

Last edited by Relentless (2016-01-25 04:12:09)

anna_gg · 2016-01-29 09:04:04

It doesn't seem to be correct

Relentless wrote:

Hi,
I found a way to fit 17 lights. There is enough space available for 18, but I would be astonished if it was usable.
I'm not really sure how to describe the (my) solution without an image. Basically:
There is a light at each corner.
Two of the corner lights opposite each other have 1.5m gaps on either side, and then there are three lights in a row 1m apart on each side (including the other corner).
There is a light as close as possible to both corner lights with a gap on either side. It looks like these lights are sqrt(2)/2 metres up/down and sqrt(2)/2 metres across from the corner lights, but that is just a guess based on physical measurement.
There is a light as far across as possible from the two corner section lights, i.e. 1m across and 1m up from one of the other corners.
Finally, there is room for two more lights as close as possible to each corner section light (those lights have a calculable distance that is about 2.95m)

Relentless · 2016-01-29 18:19:31

Why is that?

anna_gg · 2016-01-30 08:29:12

It seems we can put more...

Relentless wrote:

Why is that?

phrontister · 2016-02-01 13:37:48

Hi;

I've found 20:

Last edited by phrontister (2017-02-26 23:20:15)

Relentless · 2016-02-01 15:11:45

Well that's quite intuitive! Thank you

phrontister · 2016-02-01 22:27:35

From the way I've spaced the circles vertically, the image shows the top row of lights right at the top of the square. However, if spaced at the minimum amount there is a vacant strip of 4√0.75 across the top. Too small to do anything with, I suppose...but I did have a look to see if I could combine it somehow with other unused spaces to conjure up another spotlight, just in case this puzzle came from some sneaky puzzle setter!

For that I tried diamond shapes and equilateral triangles, but gave up after a while.

anna_gg · 2016-02-10 10:16:12

Genius!!

phrontister wrote:

From the way I've spaced the circles vertically, the image shows the top row of lights right at the top of the square. However, if spaced at the minimum amount there is a vacant strip of 4√0.75 across the top. Too small to do anything with, I suppose...but I did have a look to see if I could combine it somehow with other unused spaces to conjure up another spotlight, just in case this puzzle came from some sneaky puzzle setter!
For that I tried diamond shapes and equilateral triangles, but gave up after a while.

phrontister · 2016-02-10 11:39:18

Hi Anna;

Thanks!

However, ignore what I said about unused spaces. There are none, which is easily proved by drawing 1m-radius circles at every light.

I must have had my dunce cap on when that brainwave came to mind!

Last edited by phrontister (2016-02-10 11:40:32)

Math Is Fun Forum

#1 2016-01-23 01:26:41

Square ceiling

#2 2016-01-23 01:56:40

Re: Square ceiling

#3 2016-01-24 15:58:54

Re: Square ceiling

#4 2016-01-25 04:07:34

Re: Square ceiling

#5 2016-01-29 09:04:04

Re: Square ceiling

#6 2016-01-29 18:19:31

Re: Square ceiling

#7 2016-01-30 08:29:12

Re: Square ceiling

#8 2016-02-01 13:37:48

Re: Square ceiling

#9 2016-02-01 15:11:45

Re: Square ceiling

#10 2016-02-01 22:27:35

Re: Square ceiling

#11 2016-02-10 10:16:12

Re: Square ceiling

#12 2016-02-10 11:39:18

Re: Square ceiling

Board footer