The Problem of the Month

These problems are my attempt to give high school students, like you, a glimpse into "real mathematics" by creating a math puzzle which will challenge you to think in ways that you have never thought before. Typically, in your high school math classes, you are given a problem to solve and a fixed step-by-step procedure for solving that type of problem — follow the steps, and you are certain to get an answer. However, "real mathematics" is not like that. When mathematicians work on problems, they often start off with no idea about how to solve the problem. There are lots of math questions that no one yet knows how to solve. I invite you to stretch your mind and take my mathematics challenge. I look forward to see what you come up with — Joseph DiMuro, Assistant Professor for the Department of Math and Computer Science.

October's Math Puzzle:

Square Paths

Check out the following list of numbers:


If you add any two consecutive numbers on this list, you get a square number. For example: 5+11=16, 11+14=25, 14+2=16, and so on.

Let's call such a list of numbers a “square path”: it's a path of numbers, where any two consecutive numbers sum to a square.

Your challenge for this month: put together a square path using all the positive integers from 1 to 15. You must use only the integers from 1 to 15, and you must use each integer exactly once.

And if you manage to complete that challenge, here's a bonus challenge: put together a square path using all the positive integers from 1 to 31. Again, you may only use integers from 1 to 31, and you must use each integer exactly once.

Send your answers to this puzzle to:

Joseph DiMuro ( by October 31, 2016.


I will review your submission and whether you are correct or incorrect, I will be in touch with you to provide personal feedback, guidance, and background information regarding the problem.