Combinatorics Tiling Problem

School Name

Lucy Garret Beckham High School

Mathematics

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.

ECL 114

Start Date

3-25-2023 10:15 AM

Oral and Written

Group Project

No

COinS

Mar 25th, 10:15 AM

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.