Solange led us in an exploration of two problems – first, the Dizzy Sailor Problem and then the Perimeter of 18 Problem. The former was to challenge and deepen our own problem-solving. The latter was to have a discussion about how some of the math from the dizzy sailor connects to the perimeter of 18, which we all agreed was a problem we could do with our students.

#### The Dizzy Sailor Problem

*A dizzy sailor is standing on a 15×15 square tiled board. From their initial square they are able to move to any square sharing a common side. Due to the sailor’s dizziness, after every move they immediately make a left or right turn before repeating this process (that is, they are never able to enter and exit a square in a straight line). What is the largest number of squares the dizzy sailor can walk on if they are not allowed to repeat squares and the last step of their path must end at the square they started at?*

Before we started working on the problem, Solange had us talk about what we noticed. This way she made sure everyone understood the situation being described and all of the conditions.

Here’s what we came up with:

After working on the problem, we shared some questions of what we were wondering:

- Can the dizzy sailor walk in a perfect square?
- What is the math beyond trial and error that can help me here?
- Can the dizzy sailor walk in a rectangle? (if so, can we find the area and perimeter)
- What is the fewest number of squares the dizzy sailor can walk on?
- What if we started with a smaller grid?
- What if we all started with the same starting point?
- What if the tiled board was smaller?
- What if the tiled board had even dimensions?

Both groups came up with a similar question about using smaller boards to look for a pattern. This problem really illustrates the problem-solving strategy of solving a similar and simpler problem. This was the approach Solange used and she gave us a sheet with several grids from 2 by 2 squares to 15 by 15 to use to support our further exploration.

#### Perimeter of 18 Problem

Solange gave each group a different arrangement of squares (see docx files above) and asked the following:

*Can you add squares to this figure to make a new figure with a perimeter of 18? (Each square must share at least one complete side with another square. Trace or draw the shape that you make.)**Consider any figure made of squares where each square must share at least one complete side with another square, what is the minimum number of squares required to build a figure of perimeter of 18?**Under the same conditions, what is the maximum number of squares possible to build a figure of perimeter of 18?**What if a square didn’t have to share a complete side with another square? Would your minimums and maximums change?*

#### Problem Resources

The word docx above includes:

- The Dizzy Sailor Problem
- Blank grids (2×2 to 15×15)
- Two versions of the perimeter of 18 problem

The Supplemental Readings include:

- When Halving is Not Halving: Exploring the relationship between area and perimeter
- Maximum Area of a Rectangle with Fixed Perimeter (from Ask Dr. Math)

#### Final Thought

At the end of the meeting Solange told us that she had been thinking a lot about the kinds of math we do that challenge us as problem-solvers and the kind of problems that engage our students in productive struggle. Sometimes the same math problems can achieve both but often at CAMI we work on one kind or the other. Her goal in selecting the problems we worked on was to try and do both, so that we were all engaged in a problem and we also had some time to work on and talk about a great, open perimeter/area problem that we could use with students.

Attendance: Stephanie, Meghan, Lionel, Eric, Brian, Solange, Bree

Programs Represented: Literacy Partners, BMCC, Lehman College, CUNY Central