Plates that cover a table

[bonanova]

In our last table and plates puzzle, we filled a circular table with plates that touched without overlap.

Now we want more plates. Read on ...

I've just placed 12 [circular] plates on a rectangular table.

They don't touch each other; but they're close enough that you can't add a plate without overlap,

Edit: even tho it overhangs the edge, so long as its center is over the table.

The plates are now removed and you are asked to completely cover the table with plates.

Edit: cover means you can't see any part of the table.

Twelve won't do it, so you'll have to order some more.

But they're expensive, so you don't want to order more than are needed.

How many plates are needed to ensure the table can be completely covered?

Obviously we now allow overlap and overhang, so long as the center of the plate is on the table.

Edited February 3, 2009 by bonanova
Clarify overhang, overlap and cover.

[bonanova]

where is the proof that we can't always do better than that?

Here.

Can you sometimes do better? Yes. (4/3)x takes you from inscribed to circumscribed circles in a hex tiling.

Can you always do better? No. Not in case 2.

[Prime]

Here.

Can you sometimes do better? Yes. (4/3)x takes you from inscribed to circumscribed circles in a hex tiling.

Can you always do better? No. Not in case 2.

Sorry, I don't see the proof. I still believe, my illustration shows how 32 plates (additional 20 to the original 12) would always be enough.

Conversely, you should be able to give an example of a rectangular table and a plate, where no more than 12 non-overlapping could be placed, but 32 were not to cover the entire table. I mean specific dimensions, like table -- x*y and a plate of radius z.