Celtic Counting requires students to trace an under and over pattern – counting the Celtic loops. How many loops exist in this knot?

There are two.

How many loops does this knot have?

There are three. The following slides have larger knots, but students age 6+ should still be capable of tackling them – certainly collectively if not alone.

How many knots do square knot patterns have? These are actually easier to count because the loops are all like rectangles.

There are six.

This is a 6×7 knot. Students learning about relatively prime numbers, greatest common factors and lowest common multiples could try to come up with a general theory of how many loops are in these knots.

However, younger students can still tackle these puzzles to practice their tracing and counting skills.

This 6×7 knot has only one big loop.

This 6×8 knot has how many loops?

Just two.

This 6×9 knot has how many?

Three. Notice the symmetry.

This is just one of the loops in the previous pattern. Can the junior high student working on symmetry say anything about the symmetry of the loops in an nxm knot.

I’ll leave you with this bigger knot… Don’t go onto the next slide till you’ve solved it. Can you solve it without using paper?

There are more than two loops…

There are more than three loops…

There are four loops.

Three puzzle sheets are included in the pdf below. Here is the first.

Students should also be encouraged to create their own knots. The following sheet is a template that may help. All my knots have loops that alternate going over and under. That should be a rule for an introductory exploration.

Template to create knots…

Celtic Counting

(MathPickle, 2016)

In celebration of me going to Maths Week Ireland next week (October 15-23, 2016) here is a puzzle with a Celtic edge. Students can count the different loops (not an easy task for a six year old!) and older students can explore both symmetry of loops and can hypothesize a relationship to greatest common factors or lowest common denominations.

Here is a pdf. The first three pages are printable exercise sheets. The next page is a template for students to create their own knots. The rest of the presentation is for you to use in class as you see fit.

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!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.


Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)


Lora Saarnio