# The MOO Blog

## Math(s) and the Mosaic Frame: Competition!

17th November 2008 by Lisa

It

1. Ricky Moorhouse:

Starting with one layout of mini-frames – you have 20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 (20 factorial) = 2,432,902,008,176,640,000 ways of arranging the pictures – thats over 2 trillion and if it only takes 5 seconds for each picture, it’ll still take you 385,000 million years to work your way through the combinations. Then we need to multiply that with the number of different layouts of mini-frames within the frame, at which point we run out of digits on the calculator … simply put, it’s a product for people who like to be unique as there’s 368 million combinations for each person in the world (world population 6,602,224,175 from http://kids.yahoo.com/reference/world-factbook/country/xx–World)

2. David:

Nope, it’s angle brackets cause the trouble. Let’s try again…

First, we consider the number of ways to arrange the identical mini-frames inside the larger frame. “Tiling the plane” is a well-studied problem in mathematics and happily there is a formula for the case of tiling an MxN rectangle with 2×1 dominoes, as here.

The formula is:

2^(M*N/2) times the product of

{ cos^2(m*pi/(M+1)) + cos^2(n*pi/(N+1)) } ^ (1/4)

over all m,n in the range 0 gt m lt M+1, 0 gt n lt N+1.

Here, M = 5 and N = 8. I wrote a little script to calculate this:

product = 1;
for (n = 1; n lte 8; n++) {
for (m = 1; m lte 5; m++) {
term = ((Math.cos(Math.PI * m / 6)) ** 2 + (Math.cos(Math.PI * n / 9)) ** 2) ** 0.25
product = product * term
}
}
println product

and got 14,824.

Now, for each of these arrangements of mini-frames, in how many ways can we place the moo cards?

In the first position, we can choose 1 of 20 cards. In the second, 1 of 19. And so on. The total number of permutations is 20! (20 factorial).

Each arrangement can be hung horizontally or vertically so we multiply that by 2.

To get the final total, we multiply our two big numbers together:

2 x 20! x 14,824
= 72,130,678,738,421,022,720,000

A couple of extra assumptions I made:

– each moo card looks good vertically or horizontally but not upsidedown in either orientation.
– each moo card is included once (and only once)

3. Casey Palmer:

I concede by defeat to David, because that was exactly along the line of thought I was going with.

4. Rob:

Regardless of whether the individual cards look equally good horizontally or vertically – there is still only one best aesthetic composition of all twenty cards in the frame.

(The trick is to find it without spending 385,000 million years trying all alternative permutations first…)

5. Jon:

42, it’s the answer to everything!

6. Marjorie Stallard:

I’m going with three … I have three children, each of which would have to have(create) their own arrangement, so therefore the answer is three.

7. PJ:

I have to agree with Jon! 42 42 42!

8. Cheryl:

Math is so limiting, don’t you think? Living in a postmodern world, why limit it to one frame and 20 cards? There are as many combinations as I want. Tired of one way? Change it! Or purchase as many mosaic frames as will fill a wall. It’s dynamic, not static, right? Possibilities are endless…

9. Ellen:

How many different ways can you arrange the cards in the frame?

I’d have to say about 10 or so before my creative juices dried up and I just stopped trying.

But in mathematical terms I’m pretty sure David’s right in comment 2.

10. jay cat:

The answer is two. The (colour blind aesthetically random) way I would do it and the right way (which is how my talented aesthete spouse would do it.

11. Julia:

There are about 16000 different ways to arange minicards in Mosaic Frame. Or even more if you could not make a de?ision which card is better and could not stop ordering more and more mini cards.

12. Amy:

There are billions of ways.

My way, and your way, and his way and her way and their way and theirs, and theirs, Everyone has a different way!

13. Ralph Corderoy:

Isn’t David’s answer two times too big? The orientation of the frame, doesn’t alter the number of way the cards can be laid out. All it means is that a portrait frame, fully filled, is still legible when rotated to become landscape, and vica versa.

Looking back at the original problem, “How many different ways can you arrange the cards in the frame given the following”. The orientation of the frame doesn’t alter that.

BTW, more on domino tiling can be found on Wikipedia: http://en.wikipedia.org/wiki/Domino_tiling

14. Rain:

Your articles are for when it abostluely, positively, needs to be understood overnight.

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