Rectangles fully inside

Toss a circle of whatever radius you like onto a square grid. Color the squares that are fully inside the circle, and the squares touching or intersecting the boundary of the circle. Call the colored shape an "at least partially covered" shape. Toss another circle on the grid, but this time, only color the squares that are fully inside the circle. Call this a "fully inside" shape.

Which fully inside rectangles are achievable?


Theorem: If a uxv rectangle is achievable, then a \(u \times (v - 1)\) rectangle is achievable, so long as \(u \lt v.\)

Proof: To see this, fix a circle so its boundary touches two of the rectangle's vertices, along one of the two short sides. Now decrease the radius until one row of squares has been excluded. This can be continued until you arrive at a square of size \(u \times u.\)


Theorem: Once we've found a \(u \times v\) rectangle is unachievable, with \(u \le v,\) then \(u \times (v + 1)\) is also unachievable.

Proof: Suppose \(u\) is the height, and \(v\) the width. If we have an unachievable rectangle such that \(u \le v,\) then the polyomino must be taller than \(u.\) Increasing the radius in attempt to get \(v + 1\) will not shorten the height of this polyomino, and thus, will not return the polyomino to a height of \(u,\) as desired.


Theorem: No rectangle can have a central square. For example, a \(1 \times 3\) rectangle has a central \(1 \times 1\) square. A \(2 \times 4\) rectangle has a central \(2 \times 2\) square.

Proof: Any rectangle with a central square can be rotated \(90^\circ,\) such that the rotated rectangle is also fully inside the circle.