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Quilting squares in a math circle

Main Line Math Circle

Dec 17, 2025

Katie Haymaker

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Schedule

5:00 - 5:10 Quilt problems/Greetings
5:10 - 5:40 Introduction to the problem - 3 people and 4 people
5:40 - 6:10 Pizza
6:10 - 6:40 Discussion of the case of 5 people
6:40 - 7:00 General approach for even numbers, and open questions

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Welcome!

As you enter be sure to check in!
Sit down anywhere you’d like - but please don’t move the chairs to form large groups. (We are going to be quilting and we need a specific number of people in each group!)
Introduce yourself to group mates as they arrive.
We’ll get started soon - in the meantime give the puzzle a try!

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Thanks!!!

Support for the Main Line Math Circle is provided by the Mathematical Association of America and the Mary P. Dolciani Halloran Foundation.
Thanks to the Villanova Department of Mathematics and Statistics for the space.
Thank you to our volunteer teachers: Mr. Hawkins (director), Mrs. Hawkins, Mr. Vaccaro
Thank you to our high school mentors!
Thanks to our Villanova volunteers: Dr. Haymaker (director), Dr. Volpert, VU students
Thank you all for coming out.

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Math circle norms

Our goal is to explore fun and interesting mathematics as a community.

1) Math Circle is not about speed, but understanding; this is why you are in groups: the best way to test if you understand something is to try and explain it clearly to someone else!

2) We want everyone to have fun and to have an opportunity to think about the problems. Beware of spoilers! Ask before saying an “answer”, and give others time to think. It is ok for anyone in the group to request more time so you can think on your own before discussing.

3) We ask for you to show respect to your peers, to the high school mentors with your group, and to the college students and teachers who are here supporting the activity. Show respect to the facilitator, and please quiet all conversation when we ring the bell.

4) Share! Share supplies; share ideas; and share the floor; give others the chance to speak and contribute when you have something to add, but really think about the contributions of others!

5) If you have questions, ask anyone with a name tag.

6) Let’s treat the supplies with care and leave the space as clean as we found it.

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Introducing: The Quilt Problem

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Question 1: The original question is for a quilt-passing solution for 5 people, but let’s start small. Can we make a quilt passing pattern for 3 people that satisfies the requirements?

Question 2: What exactly are the requirements?

Each person should have one quilt to work on each week.
Every quilt should have one of each color by the end.
For any pair of people (for example, Leon and Cedar), once Leon has passed a quilt to Cedar, that does not happen again later in the quilt-passing. (But Cedar can still pass to Leon.)

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Three quilt case

Week 1: Each person has their color (PINK, BLUE, YELLOW)

Week 2: If two people were to pass to each other (like a trade), then the third person would be left without a new quilt to work on. So the passing has to be a cycle. For example:

PINK to BLUE
BLUE to YELLOW
YELLOW to PINK

Week 3: The quilt that BLUE just worked on has PINK and BLUE on it so far; it needs YELLOW. But BLUE already passed to YELLOW in Week 2, so it is impossible for that quilt to have all three colors while also satisfying the unique passing requirement.

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Take a quilting kit for your group.
Try to build a set of quilts (a passing pattern) for 4 people that satisfies all the requirements! (You’ll need to use just 4 colors and 4 sizes of square for this experiment. Set the 5th color bag aside.)
If you are done with 4 colors, work on the case of 5 quilters/colors.

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Discussion: How can we represent quilt-passing more compactly?

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Convention:

For 4 people, form a 4x4 array where the first row is the path that quilt 1 takes. Row 2 is the path that quilt 2 takes, etc.
Do this for your proposed pattern of 4 quilts!
Example: For the sub-par method with 5 quilts shown earlier, we would have:

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A Latin square is a square array with each symbol/color appearing once in row and once in each column.
Now we can rephrase the quilt-passing requirements in terms of this Latin square representation!

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Intermission

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Use the Latin square representation and the

following rules to try to make a passing

pattern for 5 people.

Rules:

Each row has one of each number 1-5.
Each column has one of each number 1-5.
Any sequence of number pairs that appears in a row, cannot appear again, ex: (1, 2) can only appear once. (The example on the left fails Rule 3 repeatedly.)

1	2	3	4	5
2	3	4	5	1
3	4	5	1	2
4	5	1	2	3
5	1	2	3	4

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Closure on the case of 5

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First quilt pass: two cases

1	2
2	3
3	4
4	5
5	1

1	2
2	1
3	4
4	5
5	3

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Filling in the other spots

1	2	4	3	5
2	3	1	5	4
3	4	?
4	5
5	1

1	2
2	1
3	4	1	2
4	5	2	1
5	3

There’s no

way to put

1 and 2 in the last row

This spot has to be 5, but then the pair 4,5

appears twice in the array.

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Filling in the other spots

(last case)

1	2	5	4	3
2	3	1	5
3	4
4	5
5	1

This spot has to be 4, but then the pair 5,4

appears twice in the array.

Putting it all together: there is no way

to fill in the 5x5 square and satisfy the

rules. So the quilt-passing scheme is

impossible for 5 people.

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Can you generalize an approach for 4 people to any even number of people?

1	2	4	3
2	3	1	4
3	4	2	1
4	1	3	2

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Example of a passing pattern for 6 people

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Row complete Latin squares

A Latin square is a square array of numbers, where each number appears once in each row, and once in column
The property that any sequence (i,j) can appear only once in a Latin square is called “Row Complete”
Our discussion just showed that if n is a positive even number, then there is a row complete Latin square of size nxn
What about other numbers??
We saw that there is no 5x5 RCLS
There is no 7x7 RCLS…
For 9?

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9x9 RCLS

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Are there RCLS for other composite numbers besides 9?

Yes!

A number n is composite if it is not prime (so it can be factored as axb where a and b are both greater than 1). There exist nxn RCLSs for any composite number n. The general answer to this question was discovered in 1996!

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Final open questions

Open questions:

Is there a row complete Latin square of size 11x11?
For any prime p>11, is there a pxp RCLS?

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Summary