## Exercise 0.2

Prove that the Fibonacci numbers count the number of different ways of inserting n dominoes into a 2 by n grid, so that each domino covers two adjacent boxes.

Solution:

Consider the following array

The first domino on the left can be vertical in which case the remaining board to be covered is one shorter

xxxxx |  |  |  |  |
xxxxx |  |  |  |  |

or it can be horizontal, in which case the only way to fill the other left square is by another horizontal domino, and the remaining board is two shorter

xxxxx | xxxxx |  |  |  |
yyyyy | yyyyy |  |  |  |

If f(n) is the number of ways of covering the 2 by n board, then we get

f(n) = f(n-1) + f(n-2), the first term counts the number of ways of finishing the job after a first vertical and the second after a first horizontal.

Exercise 0.2A Try to find a similar formula for a 3 by n board.

Notice that it can't be done at all if n is odd;

(Hint: let g(2n) be the number of ways of covering the 3 by 2n board by dominoes and let h(2n-1) be the number of ways of covering the 3 by 2n-1 board with one corner square removed. You can figure out relations among g and h as above.)