Another Kindergarten Problem

[BMAD]

I have a stamp in the shape of a circle with a diameter of 4 inches and no special design. I need to completely cover my 8.5 by 11 sheet of paper with this stamp completely. I can only use the stamp to cover the sheet, so no dipping the sheet into the ink ,

what is the least amount of times i could stamp this sheet and still completely cover it?

can you model that?

What is the most amount of times i could stamp the sheet if each successive stamp must be at least 1 inch away from the center of the previous stamp?

[bonanova]

In the second question does the 1-inch spacing refer to center-center distance?

Or must the edge of a stamp's disc be > 1 in from the center of any previous stamp?

I think you mean the latter, but just to be certain, I'm asking.

Meantime, here's the ...

The inscribed square has sides of length 2 x sqrt(2) = 2.828... inches.
8.5" x 11" then becomes 3.005 circles by 3.889 circles.

I'm guessing that by stretching them out to cover the 8.5 inches with three circles

will permit by equally slight compression vertically four circles still to cover the 11 inches.

Answer: 12 applications of the 4" stamp should cover the page.

Can I model that?

I could write the equations, but I'll defer to another solver.

[BMAD]

In the second question does the 1-inch spacing refer to center-center distance?

Or must the edge of a stamp's disc be > 1 in from the center of any previous stamp?

I think you mean the latter, but just to be certain, I'm asking.

There are three questions but the third one is meant to be the edge of the new stamp to the center of the previous circle.