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:**

Su, Francis E., et al. “Domino and Square Tilings.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**References:**

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*.

**Fun Fact suggested by:**

Arthur Benjamin