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Still Wondering... How Many Mosaics?

Still Wondering... How Many Mosaics?

In 2007, we started BLOG Zentangle and began our enjoyable series of conversations within our Zentangle community.

In reading through these blog posts with their insightful comments, we decided to bring a few of them to your attention from time to time. It is easy, for me anyway, to sometimes think of old information as stale information. But these insights and conversations are anything BUT stale!

We invite you to enjoy this post from 2018...


Begin previous post . . .


Rick writes:

I want your advice.
This post contains math. I think it’s right, but I’m not sure.
To all you math wizards (or friends of math wizards), please tell me how I did.
It’s a good story even without the math. I’d love to read your thoughts.
Thank you!                                


About a year after our Zentangle adventure began, a friend and CEO of a regional corporation invited Maria and me to give a Zentangle workshop at his company’s annual meeting for executives and managers.
We gave our workshop after their lunch and before their meeting. About 40 people were there.
The tables were arranged in a horse-shoe fashion with people sitting on the outside. We stood at the open end with a large screen behind us to project what we would draw.
We handed out paper tiles, pens and pencils and introduced the Zentangle Method. As we guided people through the process, each person became immersed in their drawing. There was pin-drop silence in the room.
When we finished, every tile was unique even though you could see how each had followed the same guidance. Each person interpreted what we said through the lens of their individual expression.
Everyone placed their finished tiles on a card table in the middle of the horseshoe. Each tile was unique even though you could see how each had followed the same guidance. Each person interpreted what we said through the lens of their individual expression.
Together, the tiles formed a “mosaic” (which is why we call them “tiles”). As people added their tiles to the mosaic, you could hear their appreciation for other’s tiles and their often surprised appreciation for their own tiles as they viewed them in this larger context.
Coffee was served and people sat back down. We gathered our gear, said our goodbyes and they proceeded with their meeting.      
We later learned that this annual meeting was the most productive and shortest in the company’s history. I like to think it was because of the card table holding their creations remained in front of everyone. The CEO’s tile was not distinguishable from the tile of the most recent hire. The mosaic that faced them was unique and could only exist because of each person’s unique tile.
After that workshop, I began thinking about a tile in a mosaic as a metaphor for an individual in a group. And I began thinking about the mosaic as a metaphor for a group.
Rearrange and rotate the tiles in different ways and you get a different mosaic.
To extend my metaphor, I wondered how many groups there might be if the same people sat in different seats with one of four different moods. So I made up a thought experiment for a group of 36 people.
Here it is.
Pretend each person has only two possible states:
  • Tired or Rested and
  • Hungry or Full
That means each person is either:
  • Rested and Full, or
  • Rested and Hungry, or
  • Tired and Full, or
  • Tired and Hungry
These options represent the four ways you can position a square tile in a mosaic.
Let’s also pretend there are six rows of six seats where the group sits.
The question for this thought experiment is:
“How many permutations can there be for a group of 36 people with two binary variables who can sit anywhere in six rows of six seats?”
Last week I decided to figure it out. When I did, I couldn’t believe the numbers! I’ll show you them at the end, but first let’s start with a smaller mosaic, one with only four square tiles. Before you scroll further, take a guess how many ways you can arrange this mosaic.

At first, I tried figure it by drawing pictures, but I soon realized that could take hours. So I searched online and ended up with this formulas for permutations of a certain number of tiles (or people) and a certain number of variables for each tile (or person):
(r^n x n!) / r
r = the number of possible rotations of a tile (3 for a 3Z and 4 for a square tile)
n = the number of tiles in the mosaic
(! represents the factorial of the number it follows. So 5! would be 5 x 4 x 3 x 2 x 1 = 120. I had to (re)learn that for this exercise.)
OK, let’s play with the four-tile mosaic.
(4^4 x 4!) / 4 = (256 x 24) / 4 = 6,144/4 = 1,536
There are 1,536 different ways you can arrange four tiles in a two by two mosaic!
Actually there are 6,144 ways you can arrange the mosaic from the perspective of any one side, but I decided to be conservative and assume that you can walk around the mosaic and view it from any side.
How many permutations do you think this three by three mosaic of nine tiles has?

Let’s do the math:
(4^9 x 9!) / 4 = (262,144 x 362,880) / 4 = 95,126,814,720/4 = 23,781,703,680
Yes, that’s over 23 billion (with a B) permutations!
Before we get to that meeting mosaic, let’s doing one more, a mosaic of 16 3Z tiles. Care to guess how many permutations of this mosaic?

Here we go . . .
(3^16 x 16!) / 3 =
(43,046,721 x 20,922,789,888,000) / 3 =
9.006574988504e20 / 3 = 3.002191662835e20 which looks like
Or, just “slightly” over 300 quintillion which is one billion times one billion times 300!
Okay, let’s solve my original question.
Any guesses? (LOL)
Here we go:
(4^36 x 36!) / 4 =
(4.72236648287e21 x 3.719933267899e41) / 4 =
1.756688818284e63 / 4 =
4.39172204571e62 or in other numbers:
or, a billion times a billion times a billion times a billion times a billion times a billion times 439,172,204.570951
Well, as I edited this I realized that if we’re talking about tiles which don’t change, but you can orient in four ways then, yes. But if we’re talking about individuals that can present themselves in four different combinations of tired/hungry, etc., then it’s that number times four.


A Zentangle mosaic is inspiring BECAUSE each tile is unique. What would be the fascination of a mosaic when each tile is the same as every other one?
What stunned me about this exercise is how MANY permutations exist for a mosaic of just a few tiles.
And when you appreciate how many more variables any individual has than the three or four variables of a tile . . . well . . .
The next time someone suggests you can presume an identity of a group based on the individuals, or the orientation of a single person because of their group, just get out 9 square tiles and your calculator and have some fun!
Another way you can approach this is to imagine four tiles and an empty 2 by 2 grid.
When you pick up the first tile you have four places in the mosaic you can place it and there are four rotations, in 90 degree increments to choose from. In other words, there are 16(4 x 4) ways to put that tile in the mosaic.
When you are ready to place the second tile in the mosaic, there are only three spaces left but you still have four ways you can orient that tile so you have 12 (3 x 4) ways to position that second tile.
With two remaining spaces for the third tile there are 8 options (2 x 4) and the last tile goes in the remaining space in one of four ways (1 x 4).
You will notice that we get the same result this way:
16 x 12 x 8 x 4 = 6,144
. . . which I divided by four for reasons stated above.

Rick Roberts


  • Art plus Math equals Awesome !!!

    Isara A on

  • Rick, all I can say is WOW! This is all mind boggling to me but I find it fascinating how you can come up with such fascinating and intriguing observations regarding the endless mosaic options. Thank you for being YOU! Now I need to go tangle! 😐

    Brenda Shaver on

  • People to the power of creativity times Zen-tiles plus Tangles = Art Ad Infinitum Thanks for giving us a place to start!

    Louise Horner on

  • RickI love your thought!thanks for your math lesson…you’re showing us that Zentangle tiles are an endless or limitless source of pleasure!

    Alicia on

  • Rick, I loved this post when you first did it and I still love it for the way it stretches my brain around the incredible potential that is revealed in creative pursuits. It is such a lovely paradox that the elegance of limits (with limited number of sides of a tile and a limited number of tiles) can be combined to create such a multitude of mosaics. It seems that the number of possible mosaics is (almost) limitless! I love having my mind boggled! Thank you.

    Leslie Hancock on

  • Loved the reading and the thought put into your reasoning.

    Now I remembered why I married a mathematician so I would not have to think that hard about a math problem.

    Marilyn I on

  • I didn’t bother to wade through your math as I’m retired and would rather spend my time tangling. But with just a cursory glance, I did see that you assumed only one side of each tile is tangled. If 2 sides are tangled, all results would be doubled. : )

    Linda Dochter on

  • I agree with Kathy Y. & I love math! 🧐🙃 I don’t think I would have attempted that calculation. But what fun!

    Josephine Wood on

  • My head hurts after reading this, Rick. I need to go decompress and do a Zentangle tile. Maybe paradox. Simple lines, can be viewed so many ways, like our mosaics!

    Ginger WhiteCZT34 on

  • Better you than me doing this math! Very interesting!

    Paulette KIrschensteiner on

  • Yeah, mine too, I’m definitely not a math nerd. It’s kinda fascinating, as it flys over my head. lol

    Alice on

  • Yes! My. Head. Exploding :)

    Wendy Beak on

  • 💝 the possibility of many options. 😃

    Rimona Gale on

  • Yes, it’s mind-blowingly complicated yet you’ve shown it is doable through math. Amazing! Thanks for sharing!

    Kim G. on

  • WOW! to the nth degree. My head is spinning….in a good way!

    Judith Shamp on

  • Hi Rick, I’m not a math wizard, but your math seems correct to me. Only you could have thought this through….what a puzzle! I’m truly amazed and impressed! Thanks for sharing this.

    Harriet Meltzer on

  • You have too much time on your hands, Rick! 😄. It’s a very interesting concept, though!

    Peg Farmer on

  • Well, you`ve certainly done some good math there, which goes a bit beyond my own education and experience. I certainly see some very high numbers, which I imagine COULD exist, but I`ve never seen them anywhere else, not even when I hear of people estimating/imagining how many stars are in the universe. I know there are `way more perms and coms than we generally think of in a daily, average way of thinking. But I still find it hard to believe that you found THAT many ways of arranging THAT few tiles! I would like to see a second mathematical opinion on this.

    Rosemary Turpin on

  • … What Kathy Y. said ☺

    Lee Kay on

  • Rick,
    Thanks, but I’ll just take your word for it! I was never very good at algebra. I’d rather use my pencil or pens to tangle my tiles, not my brain! But your “mosaic math” reminds me of the many primary, secondary, and tertiary combinations of the Meyers Briggs Type Indicator (MBTI), which could be used to explain why a group of people (or tanglers), all given the same set of directions, will interpret them so differently.

    Jessica Dykes on

  • Okay… too early in the morning for math… No comprehension without my morning routine. Could you tell me about the tile in the very lower right of the group is? LOVE it! Thanks,


    Karen Fulbright on

  • Math just validates hunches and makes it all more tangible. And delightful. Thanks for the reminder!

    Danielle DeRome on

  • This was mind boggling the first time I read it. For some reason it is even more so now that I’ve been tangling for a few years! Thanks, Rick!

    LaJuania Dorman on

  • Can you explain that again!?! 🤪

    Carol M on

  • I am a math atheist (thanks for the phrase, Calvin and Hobbs cartoon!) but found this number play fascinating! A single tile can be seen four different ways on its own. Add even just one other tile next to it and the first tile decides to pair with the second to create a new way to look at both tiles separately and together. We can rearrange our tiles hourly to give us another stimulating view of our artwork.

    Thank you for our math lesson, Rick. I think it is time for chocolate milk, shortbread cookies, and some tangling tools!

    onna on

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