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I am a teacher at a gifted center up to middle school grades, and I am looking for interesting tasks and activities for gifted 6th grade students for a course on problem solving and using varied strategies. The emphasis is on developing different skills such as creative thinking, mental flexibility, and dealing with challenges. Thanks to all the helpers.

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Justin's problem is a counting problem, and there are lots of those in any discrete math textbook. Many of those are approachable at 6th grade. Here's a free textbook that I like: https://discrete.openmathbooks.org/dmoi3.html

I also recommend geometric puzzles. Math with Bad Drawings posted a collection. Catriona Shearer makes up great ones, but they might all be too hard. I got ebooks of Geometry Snacks and More Geometry Snacks, by Vincent Pantalloni and Ed Southall. There are loads of good puzzles in those two books.

Beast Academy is a wonderful curriculum put out by Art of Problem Solving. [Disclaimer: I currently work part-time for AOPS.] BA has some pages available to print out for free. Here's a link to the level 5 printables.

I love The Art and Craft of Problem Solving, by Paul Zeitz. Most of the problems might be too hard for 6th graders, but I think you'll find a few that would work. (You might have to get that from a library. Used copies are currently priced way too high.)

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Here's a puzzle that I used to give to all grades 6th grade and up on the first day of class as something fun to do while many students were waiting to sort out schedule and laptop issues.

Draw an $10 \times 10$ 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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