Question 83
The following pieces are called $L$-triominoes.
These pieces are used to completely cover a $6 \times 6$ board. In this covering, each $L$-triomino covers exactly 3 unit squares of the $6 \times 6$ board and no unit square is covered by more than one $L$-triomino.
a) How many $L$-triominoes are used to cover a $6 \times 6$ board?
b) In a covering of the whole board, we say that a line (row or column) cuts an $L$-triomino when the line contains at least one of the unit squares covered by that $L$-triomino. If it were possible to obtain a covering of the $6 \times 6$ board in which every line cuts exactly the same number of L-triominoes, what would that number be?
c) Prove that there is no covering of the $6 \times 6$ board with $L$-triominoes in which every line cuts the same number of $L$-triominoes.
These pieces are used to completely cover a $6 \times 6$ board. In this covering, each $L$-triomino covers exactly 3 unit squares of the $6 \times 6$ board and no unit square is covered by more than one $L$-triomino.
a) How many $L$-triominoes are used to cover a $6 \times 6$ board?
b) In a covering of the whole board, we say that a line (row or column) cuts an $L$-triomino when the line contains at least one of the unit squares covered by that $L$-triomino. If it were possible to obtain a covering of the $6 \times 6$ board in which every line cuts exactly the same number of L-triominoes, what would that number be?
c) Prove that there is no covering of the $6 \times 6$ board with $L$-triominoes in which every line cuts the same number of $L$-triominoes.