Question 81
Young Abel was given a $2\times n$ board and $n$ tiles of size $2\times1$. For example, the following figure shows $n=10$: a $2\times10$ board and ten $2\times1$ tiles.
He plays by filling the board with the $n$ tiles. For $n=10$, for instance, he could fill it as follows:
There are, however, many other ways to fill the board.
(a) Calculate the total number of ways Abel can fill his board for $n=1,n=2,n=3$, that is, for boards of size $2\times1,2\times2,2\times3$.
(b) Let $a_n$ be the number of ways to fill a $2\times n$ board using $n$ tiles of size $2\times1$. Show that $a_{10}=a_9+a_8$.
(c) Calculate the total number of ways to fill the board when $n=10$.
He plays by filling the board with the $n$ tiles. For $n=10$, for instance, he could fill it as follows:
There are, however, many other ways to fill the board.
(a) Calculate the total number of ways Abel can fill his board for $n=1,n=2,n=3$, that is, for boards of size $2\times1,2\times2,2\times3$.
(b) Let $a_n$ be the number of ways to fill a $2\times n$ board using $n$ tiles of size $2\times1$. Show that $a_{10}=a_9+a_8$.
(c) Calculate the total number of ways to fill the board when $n=10$.