hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su

The Math Fun Facts App!

List All : List Recent : List Popular
About Math Fun Facts / How to Use
Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Medium level:

# Domino and Square Tilings

 Figure 1

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.

Su, Francis E., et al. "Domino and Square Tilings." Math Fun Facts. <http://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.

Keywords:    combinatorial proof, combinatorics
Subjects:    combinatorics
Level:    Medium
Fun Fact suggested by:   Arthur Benjamin
Suggestions? Use this form.
4.16
current
rating
Click to rate this Fun Fact...
*   Awesome! I totally dig it!
*   Fun enough to tell a friend!
*   Mildly interesting
*   Not really noteworthy
and see the most popular Facts!
Get the Math Fun Facts
iPhone App!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!