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Skip navigationIts Monday - and we thought we’d help jump-start your brain with a brainteaser. Back at the beginning of October, we introduced the Mosaic Frame, which made it pretty easy to transform your MiniCards in to an art piece. Based on our Flickr pool, it looks like people have already started making some early Christmas gifts and are experimenting with the teaser, without even knowing it.
Jon Walters used the Frame for his wedding photos:
While Yaelfran has used it for her artwork:
Now, let’s get back to the Brain Teaser. It all started when Paul (our product designer) first showed me the design, my brain started to hurt. Why? Well, I kept trying to calculate how many different ways I could put together a set of cards in a frame. I must admit my University Math class was ages ago. So, can you help me out?
The Challenge:
How many different ways can you arrange the cards in the frame given the following:
1. you have 20 different MiniCards
2. each MiniCard looks equally good vertically or horizontally
3. each MiniCard must be included in the frame arrangement only one time
Remember, the frame can also hang vertically or horizontally (perhaps that adds another 2 variations?) Here’s a little video we made to show you how it works. It should give you an idea of how the little frames fit in to the bigger frame…
So, if you think you know the answer, leave your answer and calculations in our comments. Based on collective understanding here at MOO we will pick the winner from the selection of answers we receive by end of week. The winner will receive a Mosaic Frame for free. (Your answer doesn’t have to be right but needs to be well thought out - remember ‘Part Marks’ ). We’ll announce the winner next Monday.
Good Luck!
PS If you want tips on how to put together the frame Paul has provided his expert advice!
13 Responses to “Math(s) and the Mosaic Frame: Competition!”
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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)
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)
I concede by defeat to David, because that was exactly along the line of thought I was going with.
The answer is one.
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…)
42, it’s the answer to everything! :-)
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. :)
I have to agree with Jon! 42 42 42! :D
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…
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.
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.
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.
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!
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