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riddles >> hard >> Tiling a circular hall with circular tiles
(Message started by: JocK on Jul 24th, 2004, 12:48pm)

Title: Tiling a circular hall with circular tiles
Post by JocK on Jul 24th, 2004, 12:48pm
You need to tile a circular hall. You have contacted a company that can construct circular tiles of any specified diameter up to giant-sized tiles with diameter equal to the radius of your hall. The sales manager of this company wants to sell a lot of tiles and makes you a special offer that is only valid in case you ensure that you cover at least 90.7% (the hexagonal packing fraction = sqrt(3)[pi]/6) of your hall with their tiles. Expecting that this requirement will cause you to order many identical tiny tiles, he offers to deliver the tiles for a fixed price of 1$ per tile regardless of the size, even if each tile has a unique specified size.

You are very tight on budget, and you want to minimize your cost. Based on the above offer, what minimum amount do you need to pay the tile company to ensure you can tile your hall? What are the sizes of the tiles you are going to order? Are you sure one cannot tile the hall with one less tile?

J8)CK

PS. You can't break the tiles, and also the tiles are not allowed to overlap!

Title: Re: Tiling a circular hall with circular tiles
Post by Jack Huizenga on Jul 24th, 2004, 2:55pm
Sorry, but this ridde doesn't make any sense.  You haven't attributed a cost to the tiles besides the delivery cost, and you haven't given a reason why the "one big tile" solution isn't optimal...

Title: Re: Tiling a circular hall with circular tiles
Post by JocK on Jul 24th, 2004, 4:48pm

on 07/24/04 at 14:55:07, Jack Huizenga wrote:
Sorry, but this ridde doesn't make any sense.  You haven't attributed a cost to the tiles besides the delivery cost,

??



on 07/24/04 at 14:55:07, Jack Huizenga wrote:
and you haven't given a reason why the "one big tile" solution isn't optimal...

It says: "any specified diameter up to giant-sized tiles with diameter equal to the radius of your hall". So one tile would  cover only 25% of the hall. Not even close to the required ~ 90.7%.


If the story is difficult to follow; forget about it:

What is the minimum number of non-overlapping circular disks that you need to fit inside a circle so as to fill it to at least hexagonal close-packing density (sqrt(3)[pi]/6), under the constraint that none of the disks can have a diameter exceeding the circle radius.

J8)CK

Title: Re: Tiling a circular hall with circular tiles
Post by Jack Huizenga on Jul 24th, 2004, 4:57pm
Oh ok sorry about that, I misread the question.

Title: Re: Tiling a circular hall with circular tiles
Post by SWF on Jul 26th, 2004, 9:41pm
After trying a couple of possibilities, best I could find so far is 20:
[hide]Start with three mutually tangent circles of same diameter that all are inside and tangent to the bigger circle.  Then keep adding the largest circle that fits in the remaining space until 20 circles[/hide].

Title: Re: Tiling a circular hall with circular tiles
Post by JocK on Jul 27th, 2004, 1:04pm
If I'm not mistaken, the 20 biggest cicrles of this Apollonian packing together exceed 90 % packing density, but they don't reach the 90.69 % hexagonal packing density. Even for 22 circles only 90.67 % is reached (still short by 0.02%...!).

This all is relevant as the best result I found so far is [hide]22[/hide] circles.

J8)CK

Title: Re: Tiling a circular hall with circular tiles
Post by Virat Agarwal on Oct 7th, 2004, 5:22am
Yes there is also available the circular tiles in natural stones which  can be interlocked and can be laid down to any area on the floor.these tiles are developed in the natural stones like sandstones,limestones and granites

Title: Re: Tiling a circular hall with circular tiles
Post by John_Gaughan on Oct 7th, 2004, 5:39am

on 10/07/04 at 05:22:21, Virat Agarwal wrote:
Yes there is also available the circular tiles in natural stones which  can be interlocked and can be laid down to any area on the floor.these tiles are developed in the natural stones like sandstones,limestones and granites

Nonoverlapping circles can only touch at a tangent point, leaving space around that point. Interlocking is out of the question. Polygons, especially regular polygons such as squares and hexagons, can interlock quite well.



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