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How Many Mosaics?

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


  • hmm…don’t know…don’t care…just Tangle!

    Jeanne Hertzel on

  • I loathe math and totally love Zentangle. All of this fixation with the crazy numbers is way beyond me. I think I will go appreciate my last Zentangle tile and also create a new one. That is two in my world and I love my world. Thank You.

    Victoria Smith on

  • @Lisa Locke
    When you assemble a mosaic of square tiles on a table, is it a different mosaic if you look at it from the other side of the table?

    I decided the answer should be, “no” for both practicality and to be as conservative as possible.

    If you could live long enough to arrange the mosaic in its full set of permutations, and if you had an eidetic memory, you would notice that one quarter (in the case of square tiles) of the permutations reappeared in rotations of 90, 180, and 270 degrees . . . in other words, as if you viewed it from different sides of a square table.

    Therefore I divided by r.

    RIck Roberts on

  • I’m afraid ya’all cannot look to me for the math verification. It’s all above my head, even if only two tiles were used. That said, it IS a good story, regardless of the math understanding.

    LadyD on

  • I came up with 256 permutations for four tiles. Each tile can be in one of four positions. So, that is 4 tiles x 4 positions = 16 permutations. Each tile can be turned 4 ways. So, that is Side 1 + Side 2 + Side 3 + Side 4 for a total of four possible directions each tile can face. So, it would be 4 positions x 4 sides = 16 possible permutations again. And, 16 possible positions x 16 possible directions = 256 possible permutations for four tiles with four sides. I dropped this math course though and changed to trigonometry. It was easier :) I hope you find a math genius to solve this for us.

    Pixel Ghost on

  • Hi Rick,

    I think your math needs revisiting. 4 tiles would be 4!, which is 4×3×2x1=12.
    The combinations would be:
    1 2 1 2 1 3 1 3
    3 4 4 3 2 4 4 2

    1 4 1 4
    2 3 3 2
    That’s six combinations with tile one in upper left. Four tiles x 6 is 24.
    Nine tiles would be:
    9×8×7x6x5×4×3x2x1 = 362,880. Hope this helps.
    I love Zentangle and love math.

    Gail Minichiello on

  • I quickly got (r^n x n!), but I don’t understand why you’re dividing by r.

    Lisa Louque on

  • Heavens To Murgatroyd! So when are you taking this to the United Nations?

    Jean Mackie on

  • Thanks for all the comments! The math hasn’t been independently verified yet. But the newsletter that’s coming out will link to this blog. I’m looking forward to either verification OR correction.

    Rick Roberts on

  • So……..

    Did the mathematics ever get confirmed?

    Laura Story on

  • Oh Rick….this is SO wonderful! I’m going to forward this on to my son, a Phi Beta graduate with his degree in math. He loves this stuff! We’ll have to see if your numbers sync! Thanks for the post….my inner geek loves this stuff.

    Rita miller on

  • I never thought about all the possible variations of a mosaic like this before…

    Jackie on

  • This is sooooo cool. I love this stuff. My husband and I just calculated everything and corrected each other. This is soo great. Thank you!

    Tanja Francke on

  • Wow, I guess I have been out if school too long, because I get 40 for the original problem of 4 tiles on a 2×2 grid. Back to tangling…

    Dana "Jonesy" Jones on

  • There is a character in the book The Phantom Tollbooth called the “Mathemagician.” That’s you, Rick! Thank you for sharing the marvelous infinite number of possibilities that Zentangle offers to all of us!

    Jane Rhea on

  • Rick, you blew my mind with this post. Math was never one of my strong points and with only one right answer, it frustrated me. I love the simplicity of the Zentangle method with NO wrong and unimaginable possibilities.

    Lianne Woods on

  • My first Physics professor thought frequently about problems like this, and you remind me of him. I love this type of puzzle and truly appreciate the time and energy and thought process involved. You have a unique way of humbling us, reminding us that we are a part of something so much greater, yet we all have a place in that vastness.

    Terri Delaune on

  • Even really big numbers have names, so 4.39e62 is also known as 439 Novemdecillion! I love zentangle and maths, but sometimes I forget how much overlap there is between maths and art. Surely numbers are language that we use to create all our tangles?They define the geometry, symmetry and patterns of our tangling, so when you think you are just drawing, you are actually, subconsciously doing maths! The next question surely is, if Cris’ s daughter had those 36 tiles to rearrange in all those permutations, and each new arrangement took 1 second, how many years would it take her to complete all of them?

    Lucy Farran on

  • That reminds me of something I heard years ago. If there are two people in a room, there are really six – who I think I am, who you think I am and who I really am. If there are 12 people in the room, there are really 156 people. It’s a wonder we ever really understand each other

    Paddy Brown on

  • Ok i was all raring to go in solving this problem and read it twice before i finally said, in solving for the problem, what info is this giving to the audience, and what can they do with it once they know this? Im still struggling with understanding that first. Can you tell i was a CFO in food ingriedients industry and a CAO/COO for large hospital during my career?! ?

    Matt Wieczkowski, CZT , NJ on

  • Well, very interesting…. It will take me some years to figure it all out…. so back to tangling instead. Leaving the maths to people who know about maths. There is a place for everybody ;0)

    Karin CZT Belgium on

  • ?this is exactly why I draw instead of do math…holy guacamole…☺️, but truly quite fascinating Rick..thank you.

    Jody Genovese on

  • It is very interesting! Thank you Rick! ^^

    Yoonmi on

  • Loooove this!!! Great going back to calculating chances for fun ? it’s been almost 30 yrs ago when I had these kind of calculations in my math lessons.

    Love it still!!

    ArjadL on

  • The arithmetic is awesome but most awesome is the effect art has on humans. That wee bit of time a person spends tangling a tile puts the right side of their brain into a whole different perspective. It’s no wonder people perform tasks more efficiently after tangling. It’s like doing your stretches before running. Nice blog topic.

    Mary Kay Watson on

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