Stirring Paint
(MathPickle, 2020)
Above is a solution to a stirring paint puzzle. Can you guess the rules?
#1 When the puzzle is solved, each colour forms exactly one loop.
Example: if you start with the blue in the upper left, you move one space to a blue arrow pointing 2 down. Going 2 down, you find a blue arrow pointing 1 to the right. From there, you move 2 up and then 2 to the left, back to your starting blue arrow.
All the blues are part of this single blue loop. The yellow and red are part of smaller loops.
#2 Together, all the loops must cover the entire canvas.
Thanks to Steve Heller, you can find a lot of Stirring Paint Puzzles here.
Let’s look at how to solve one of the smaller ones…
Clues come in different forms. Sometimes, like the blue square at the top, we do not know the direction or the number. Sometimes, like the gray square labelled “2” at the top right, we know the number, but not the colour or direction.
Steve Heller wants students to make logical deductions one step at a time.
Example: What direction is the 2-square at the bottom middle? Follow the steps one at a time to the solution:
The right image above is part-way to the solution. Which arrow points to the upper right 2-arrow?
Every arrow must have one “parent” that points to it, and one “child” which it points to.
All the yellows must be part of the same loop.
And we have our solution!
Here are some other stirring paint puzzles. You’ll find a lot more on the link above.
Standards for Mathematical Practice
MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:
MP1 Toughen up!
Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.
MP2 Think abstractly!
Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.
MP3 Work together!
Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. Try pairing up elementary students and getting older students to work in threes.
MP4 Model reality!
Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.
MP5 Use the right tools!
Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.
MP6 Be precise!
Students learn to communicate using precise terminology. Students should not only use the precise terms of others but invent and rigorously define their own terms.
MP7 Be observant!
Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist! 😉
MP8 Be lazy!?!
Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?
Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.
Gordon Hamilton
(MMath, PhD)
