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 ≥ 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 ≥ 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.
