A challenging number pattern problem

My younger son struggled with a number pattern problem from an old MOEMS test today. I enjoyed talking through it with him tonight because it was interesting to see how he approached the pattern in the numbers once he saw it – his approach was quite a bit different that what I was expecting.

Here’s an introduction to the problem and our initial talk that gets us on the path that surprised me:

So, my surprise in the last video is that he wanted to go to the end of a row and subtract a certain amount to get back to the beginning. I thought it would be interesting to see if he could see that you could also add 1 to the square at the end of the last row. This idea was hard for him to see, but eventually we got there.

At the end of the last video we talked about how the odd numbers relate to the perfect squares. The sequence of rows in the original problem hints at the relationship, though for me, at least, the connection doesn’t jump off the page. To get a better sense of that relationship we went to our kitchen table and looked at the relationship using snap cubes:

So, a fun little project starting from an old math contest problem. Ultimately the lesson I’m hoping to convey with my son here is about looking for patterns. The connection between arithmetic and geometry in the last part is also something that I hope he finds interesting. I always find it fun when geometry helps us understand arithmetic a little better.