Playing with Triangular Numbers

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 \u3e 4 where our problem has no solution?” So in our work we proved the following theorem, derived from McMullen’s question: Theorem: Let k \u3e 4 be a square. Then there exist k consecutive triangular numbers that add up to a triangular number