I had dinner with the kiddos at Round Table tonight. (Side note: yum.)
And then I asked Ainsley (4): “How many squares do you see?”
She counted the top row: “1… 2… 3… 4…” Then a long pause, followed by (pointing along the bottom row): “5… 6… 7… 8…”
We both stared at the receipt for a little while longer. Neither of us spoke for a bit. Then—for better or for worse—I broke the silence, tracing the perimeter of a larger square consisting of a 2-by-2 array of smaller squares. She helped me count this 9th square plus two more just like it, and we landed on a total of 11.
It’s hard to tell if those last three squares were lost on her, even after we traced them together. In all likelihood, they probably were. Anyway, that’s not the point of the post. (Nor is her original counting of eight, two rows of four at a time, though I’m confident there’s more than enough material for a blog post in the “how do you see it” conversation.)
A Familiar Problem
Here’s where I am headed with this.
I found myself recalling the classic problem “how many squares on a chessboard?” And then I wondered, “how many squares on an n-by-n board?” I’ve explored both of these questions before, and while I cannot recall the generalization off the top of my head, I’m confident I could find an explicit formula if I tried.
A New (to me) Problem
That sequence of thought led to a new question, or rather, a question that’s likely been posed many times by others but is brand new to me:
How many squares on an m-by-n board?
I’m sharing it here sans answer because I don’t yet have an answer (and it’d be no fun to spoil your fun in tracking one down). But I’m excited to start exploring. It may not be the sexiest or most challenging problem in the world, but it’s grabbed my attention nonetheless.
And if I’ve piqued your interest, I’d love if you gave it a try as well.
In which case, drop a line in the comments describing your approach, your answer, and (bonus!) another question about rectangles/squares/boards that interests you.