BdMO Online Forum

Induction, twice! One for obvious reason. Second (not really induction, but explicit construction) for proving that every L-shaped triomino like figure with with $2^k\times 2^k$ squares (in the place of the unit squares in normal triomino) can be covered just using triominos.

firstly, let us prove that if the missing square is at the corner, what happens...the $2\times 2$ square it is situated in is nothing but a trimino.let us take a trimino at it's corner so, the rest part is three triminos.because, we have taken the three unit squares from three $2\times 2$ squares.so, this $4\times 4$ is filled with triminos.the base case is complete.let us think about the surrounding three $4\times 4$ now we can take a trimino using three unit squares from these three $4\times 4$ squares like the base case the rest of them will be filled with triminos.by induction we can prove that for any $2^k\times 2^k$ square.where the corner square is missing.

let us now prove that in any $2^k \times 2^k$ square the missing square can be moved to the corner.
we can divide the square in $4$ parts any of them consists the missing piece.we can rotate it to make the missing piece at the corner as much as possible.if it is not at the corner.we can divide it in $4$ parts again and again and rotate again and again.

no matter how much change we do, the first part does not change.so it is done i think.