Exercise 0.2

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