» Required Reading » Table of Contents » Chapter 3: Vectors, Dot Products, Matrix Multiplication and Distance » Section 0.2 Exercise 0.2
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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;

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:

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.