How many ways can you tile a (1 x n) board with (1 x 1) squares and (1 x 2) dominoes?
Let's see…
A (1 x 1) board can only be tiled 1 way.
A (1 x 2) board can be tiled 2 ways: either two squares or one domino.
A (1 x 3) board can be tiled 3 ways: three squares, or a domino-square, or a square-domino.
A (1 x 4) board can be tiled 5 ways…
Hey, they are all Fibonacci numbers! These are numbers you get when you start a sequence 0 and 1 and generate the next sequence member by adding the previous two numbers: i.e., 0, 1, 1, 2, 3, 5, 8, …
Presentation Suggestions:
Draw pictures, and have students count the number of tilings themselves.
The Math Behind the Fact:
The Fibonacci property can be seen by conditioning on whether the first tile is a square or domino: the number of tilings where the first tile is a square is the number of ways to tile the remaining (n-1)-length board, and the number of tilings where the first tile is a domino is the number of ways to tile the remaining (n-2)-length board. A generalization of this idea can be used to develop combinatorial interpretations for many “Gibonacci” identities (recurrences in which the first terms are not necessarily 1 and 1); see the Benjamin-Quinn-Su reference for more on this.
An alternate generalization asks how many ways to tile an (m x n) board with dominoes; note that the (2 x n) case is equivalent to tiling a (1 x n) board with squares and dominoes. See the Graham-Knuth-Patashnik reference.
other generalizations can be found in the Brigham-Caron-Chinn-Grimaldi reference. If you like this Fun Fact, you'll like all the stuff that is in the Benjamin-Quinn book in the references.
How to Cite this Page:
References: Fun Fact suggested by:
Su, Francis E., et al. “Domino and Square Tilings.” Math Fun Facts.
A. Benjamin, J. Quinn, and F. Su, “Phased Tilings and Generalized Fibonacci Identities”, Fibonacci Quarterly, 2000.
Brigham, Caron, Chinn and Grimaldi, “A Tiling Scheme for the Fibonacci Numbers”, J. Recreational Math, 28(1), 1996-7, pp. 10-17.
Graham, Knuth, Patashnik, Concrete Mathematics, Section 7.1. Benjamin and Quinn, Proofs that Really Count
Arthur Benjamin