# My Go-To Math Riddle: How Many Squares are in a 10 x 10 Grid?

by Justin Skycak on

Q: Draw a 10 x 10 square grid. How many squares are there in total? Not just 1 x 1 squares, but also 2 x 2 squares, 3 x 3 squares, and so on. A: The total number of square shapes is the total sum of square numbers 1 + 4 + 9 + 16 + ... + 100.

Cross-posted from here.

Here’s a puzzle that I used to give to all grades Algebra through Calculus on the first day of class as something fun to do while many students were waiting to sort out schedule and laptop issues.

Draw a $10 \times 10$ square grid. How many squares are there in total? Not just $1 \times 1$ squares, but also $2 \times 2$ squares, $3 \times 3$ squares, and so on.

I liked this puzzle in particular because the solution was not immediately obvious to even the brightest / oldest students, yet all students (even the younger grades / weaker students) were able to understand the goal of the problem and work through concrete examples to notice a pattern that they could extrapolate to find the solution. (The higher grades / brighter students just picked up on the pattern faster.)

Whenever students needed some guidance getting started, I would explicitly walk them through the process of drawing up a $3 \times 3$ grid and counting the number of $1 \times 1$ squares (there are $9$), $2 \times 2$ squares (there are $4$), and $3 \times 3$ squares (there is $1$), for a total of $9 + 4 + 1 = 14$ squares. I would then leave them alone to do this for a $4 \times 4$ grid and then a $5 \times 5$ grid, after which I would help them notice the following pattern:

\begin{align*} 3 \times 3 \textrm{ grid} \quad \to & \quad 9 + 4 + 1 \\ & \quad 3^2 + 2^2 + 1^2 \\[5pt] 4 \times 4 \textrm{ grid} \quad \to & \quad 16 + 9 + 4 + 1 \\ & \quad 4^2 + 3^2 + 2^2 + 1^2 \\[5pt] 5 \times 5 \textrm{ grid} \quad \to & \quad 25 + 16 + 9 + 4 + 1 \\ & \quad 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\[5pt] \end{align*}

This pattern is almost a “punch line”: What’s the total number of squares (square shapes)? Well, it’s the total sum of squares (square numbers).

Extrapolating the pattern to solve the puzzle:

\begin{align*} 10 \times 10 \textrm{ grid} \quad \to & \quad 10^2 + 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\ & \quad = 385 \textrm{ squares in total} \\[5pt] \end{align*}

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