#### Title

Combinatorics Tiling Problem

#### School Name

Lucy Garret Beckham High School

#### Grade Level

12th Grade

#### Presentation Topic

Mathematics

#### Presentation Type

Mentored

#### Abstract

Combinatorics is the field of mathematics studying the combination and permutation of sets of elements and the relationships that constitute their properties. A problem proposed in The College Mathematics Journal asks for a closed form expression for the number of ways to tile an n x n square with 1 x 1 squares and (n - 1) x 1 rectangles (each of which may be placed horizontally or vertically) for an integer n â‰¥ 3. Using a 3 x 3 square as a starting point, we determined all of the possible cases by hand. Upon doing so, we were able to determine generalizable patterns for n cases and formulate combinations for each. Finally, using a series of identities we rewrote the formula into a closed form.

#### Recommended Citation

Matheson, Peyton, "Combinatorics Tiling Problem" (2023). *South Carolina Junior Academy of Science*. 95.

https://scholarexchange.furman.edu/scjas/2023/all/95

#### Location

ECL 114

#### Start Date

3-25-2023 10:15 AM

#### Presentation Format

Oral and Written

#### Group Project

No

Combinatorics Tiling Problem

ECL 114

Combinatorics is the field of mathematics studying the combination and permutation of sets of elements and the relationships that constitute their properties. A problem proposed in The College Mathematics Journal asks for a closed form expression for the number of ways to tile an n x n square with 1 x 1 squares and (n - 1) x 1 rectangles (each of which may be placed horizontally or vertically) for an integer n â‰¥ 3. Using a 3 x 3 square as a starting point, we determined all of the possible cases by hand. Upon doing so, we were able to determine generalizable patterns for n cases and formulate combinations for each. Finally, using a series of identities we rewrote the formula into a closed form.