[psyang]

Seems things have died down on this problem. Here's the solution.

Spoiler!

Let's say the coins are circles of radius 1.

Take one coin on the table. If we draw a circle of radius 2 around that coin's centre, we can guarantee that no other coin's centre is inside that circle (if it were, the coins would be overlapping).

Let's draw circles of radius 2 on all of the coins. Let's assume that these new circles do not cover the table. That means that there exists some space that is not covered by any of the circles of radius 2.

But if that were the case, then we could place a new non-overlapping coin of radius 1 with its centre in that uncovered space, which is a contradiction.

Therefore, the circles of radius 2 completely cover the table.

Let's scale the table and circles down by 1/2 in the x and y direction. The table is now 1/4 of the size of the original table, and the circles of radius 2 are now circles of radius 1 which can be considered coins. We now have 100 coins covering a table 1/4 the size of the original based on the original coin configuration.

We can therefore cover the entire table with 400 coins.