A single-square-covered board is a board of 2" x 2" squares in which one square is covered
with a single square tile. Show that it is possible to cover the this board with triominoes
without overlap
Answers
2g of Ferrous sulphate crystals are heated in a dry boiling tube .
list any two observations
Name the type of chemical reaction taking place
write the chemical equations for the reaction.
size of the board is 2nn x 2n
Number of squares = 2n x 2n = 4n
Number of squares covered = 1
Number of squares to be covered = 4n – 1
4n – 1 is a multiple of 3
Case 1 : n = 1
The size of the board 2 x 2
one triominoe can cover 3 squares without overlap.
Case 2 : n ≥ 2
1. place a triominoe at the center of the entire board so as to not cover the covered sub – board.
2. One square in the board is covered by a tile. The board has 4 sub – boards of size 2 x 2 . 2n – 1 2n – 1.
Out of 4 sub – boards one sub – board is a single square covered sub – board
One triominoe can cover remaining three sub –boards into single square covered sub – board. The problem of size n is divided into 4 sub – problems of size (n – 1). Each sub – board has 2n-1 x 22n-1 – 1 = 22n-2 – 1 = 4n – 1 squares to be covered.
4n-1 – 1 is also a multiple of 3
In this, the 2n x 2n board is reduced to boards of size 2×2 having are square covered. A triominoe can be placed in each of these boards and hence the whole original 2n x 2n . board is covered with triominoe with out overlap.