## 2019 - 22nd Annual Steven Galovich Memorial Student Symposium

#### Presentation Title

Playing with Triangular Numbers

#### Department or Major

Mathematics

#### Location

Lillard 132

#### Abstract

A triangular number is a number *N* that satisfies that *N* dots can be arranged in increasing order to form an equilateral triangle. A triangular number *N* can be represented in the form *N* = ^{n(n+1)}⁄_{2} , for some positive integer *n*.

In the article “Playing With Blocks”, McMullen noticed that sometimes, there are consecutive sequences of triangular numbers that add up to form another triangular number. For example 1 + 3 + 6 = 10, where *k* is the number of triangular numbers added. He asked the following question: “Every value of *k* except *k* = 4 that I looked at yields at least one valid solution. Is there a *k* > 4 where our problem has no solution?” So in our work we proved the following theorem, derived from McMullen’s question:

Theorem: Let *k* > 4 be a square. Then there exist *k* consecutive triangular numbers that add up to a triangular number.

#### Presentation Type

Individual Presentation

#### Start Date

4-9-2019 10:30 AM

#### End Date

4-9-2019 11:45 AM

#### Panel

Research that Counts: Applied Insights and Theoretical Provocations

#### Panel Moderator

DeJuran Richardson

#### Field of Study for Presentation

Mathematics

Playing with Triangular Numbers

Lillard 132

A triangular number is a number *N* that satisfies that *N* dots can be arranged in increasing order to form an equilateral triangle. A triangular number *N* can be represented in the form *N* = ^{n(n+1)}⁄_{2} , for some positive integer *n*.

In the article “Playing With Blocks”, McMullen noticed that sometimes, there are consecutive sequences of triangular numbers that add up to form another triangular number. For example 1 + 3 + 6 = 10, where *k* is the number of triangular numbers added. He asked the following question: “Every value of *k* except *k* = 4 that I looked at yields at least one valid solution. Is there a *k* > 4 where our problem has no solution?” So in our work we proved the following theorem, derived from McMullen’s question:

Theorem: Let *k* > 4 be a square. Then there exist *k* consecutive triangular numbers that add up to a triangular number.