Question 68
Define a size-$n$ tromino to be the shape you get when you remove one quadrant from a $2n \times 2n$ square. In the figure below, a size-1 tromino is on the left and a size-2 tromino is on the right.
We say that a shape can be tiled with size-1 trominos if we can cover the entire area of the shape—and no excess area—with non-overlapping size-1 trominos. For example, a $2 \times 3$ rectangle can be tiled with size-1 trominos as shown below, but a $3 \times 3$ square cannot be tiled with size-1 trominos.
a) Can a size-5 tromino be tiled by size-1 trominos?
b) Can a size-2013 tromino be tiled by size-1 trominos?
Justify your answers.
We say that a shape can be tiled with size-1 trominos if we can cover the entire area of the shape—and no excess area—with non-overlapping size-1 trominos. For example, a $2 \times 3$ rectangle can be tiled with size-1 trominos as shown below, but a $3 \times 3$ square cannot be tiled with size-1 trominos.
a) Can a size-5 tromino be tiled by size-1 trominos?
b) Can a size-2013 tromino be tiled by size-1 trominos?
Justify your answers.