# An Uberlord from the world of football

Ok, this isn’t strictly Statistics, but I came across it while researching for another post and it seemed fun, so I thought I’d share it.

JiGaZo is a 300-piece jigsaw puzzle with a difference. The pieces are identically-shaped, 90-degree rotationally symmetric, sepia-coloured (to different degrees of shading) and have a colour-coded symbol on the back. You take a picture of yourself or anyone else. You upload that picture to your computer, run it through the software provided, and it spits out a grid of the codes on the back of the jigsaw pieces. You then construct the jigsaw following those coded instructions, and the result is a jigsaw reproduction of the image you uploaded.

Perhaps it’s better explained by the following ad:

You can get JiGaZo for around a tenner at Amazon. Just picture your loved one’s face on Christmas Day when they realise that not only have you constructed a 300-piece jigsaw of them as a present, but they can also disassemble it and return the favour to you in time for Boxing Day.

Christmas shopping ideas: just one of the services provided by Smartodds loves Statistics.

Now, to give this post some relevance to Statistics, we might ask how many unique images can be made with a JiGaZo set? These are the types of calculations we often have to make when enumerating probabilities in all sorts of statistical problems.

Have a quick guess at what the answer might be before scrolling down…

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So, first we can calculate the number of ways the 300 pieces can be placed in the grid. There are 300 choices for the first piece; that leaves 299 for the second; then 298 for the third; and so on. Since each of these choices combines with each of the others, the total number of such arrangements is

$300 \times 299 \times 298 \times .... \times 3 \times 2 \times 1$

By convention this is written 300! and it’s HUGE: approximately the number 3 followed by 614 zeros.

To put that in perspective, it’s believed the number of stars in the observable universe is ‘only’ 1 followed by 21 zeros. So, if every star had its own universe, and every star in that universe had its own universe, and every star in that universe had its own universe and we kept doing that – putting a universe on every new star – a total of around 29 times, then we’d have about 300! stars in total.

But that’s not all. For every one of those 300! arrangements of the Jigazo pieces, each piece can be rotated in 4 different ways. So we have to multiply 300! by 4 three hundred times. Finally we divide that answer by 2 since any arrangement is arguably the same if it’s upside-down – I can just rotate it and get the same picture.

So, the final answer is

$300! \times 4^{300} /2$,

which is roughly 6 followed by 794 zeros. So, apply that universe on every star procedure another 8 times or so, and you get close to the number of unique JiGaZo images.

Since even the fastest computer in the world would take much longer than the age of the universe to run through all of those possibilities, you start to realise that the software that comes with JiGaZo, which aims to find a pretty good match for any input, must be a smart piece of image mapping.

But crucially… how well does it work? I guess the answer to that is determined by how easily you can recognise the following Uberlord from the world of football…