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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.)

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