MathDB

1. Can a

7 \times 7~

square be tiled with the two types of tiles shown in the figure? (Tiles can be rotated and reflected but cannot overlap or be broken)
2. Find the least number

N

of tiles of type

A

that must be used in the tiling of a

1011 \times 1011

square. Give an example of a tiling that contains exactly

N

tiles of type

A

. [asy] size(4cm, 0); pair a = (-10,0), b = (0, 0), c = (10, 0), d = (20, 0), e = (20, 10), f = (10, 10), g = (0, 10), h = (0, 20), ii = (-10, 20), j = (-10, 10); draw(a--b--c--f--g--h--ii--cycle); draw(g--b); draw(j--g); draw(f--c); draw((30, 0)--(30, 20)--(50,20)--(50,0)--cycle); draw((40,20)--(40,0)); draw((30,10)--(50,10)); label((0,0), "

(A)

", S); label((40,0), "

(B)

", S); [/asy] Proposed by Muralidharan Somasundaran

Problem Statement