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## Jigazo Puzzle – 300 Pieces Make Billions of Faces

The Jigazo puzzle – the new thing out of Japan – is a puzzle of 300 pieces, all of the same shape, in a square 15 pieces wide by 20 pieces high. Every piece on it is the same color, in varying degrees of intensity and degree. Parts are marked with unique icons. These icons allow the pieces to be individually identified, so that they can be placed in the correct position to create an image by following the image map for the desired image. By arranging these pieces in just the right way, almost any image can be recreated.

In Japan, the word Jigazo means “self-image”. To create a self-portrait (or any other photo of your choice) with the Jigazo puzzle, just email a copy of your photo (or any other photo) to the puzzle maker, and within minutes, you’ll get a map. This map shows where each of the 300 pieces must be placed, and the correct orientation of each piece, to form the finished image. Of course, there is a limit to the detail that the Jigazo puzzle can reproduce – but the fact that it works at all is incredible!

OK, so now we know how a bunch of pieces of the same shape but different colors can be moved around to make different pictures – but how is it possible that only 300 pieces can make a picture of a person on Earth? create? After all, there are about 7,000,000,000 people on earth – surely one puzzle can’t make that many different pictures… can it?

Yes, he can – without trying! In fact, the number of different images that this puzzle can create is imagination. The total is a number so large that it exceeds the number corresponding to anything real in the known Universe!

Let’s see how it is possible: Start with an arbitrary arrangement of 300 pieces in the puzzle. That’s picture number one. Now, since all the pieces have similar shapes, each of the 300 pieces can be placed in four different positions, rotating 90 degrees each time. To do this with the piece in the upper left corner, we will create four (also small) different images.

Now, in all four versions of the image, we can take the next piece on the top row, and rotate it to four different positions. This means that each of the four (very few) different images we created by rotating the first part now has four different versions as well.

Now, you can see a pattern emerging. By rotating the first part, we have 4 different images. Turning the second slice creates 4 more pictures for each of those 4 pictures. Therefore, for the first 2 parts, the total number of images is given by 4 x 4 = 16. This can also be written as a concrete formula: 4^2 = 4 x 4 = 16. In this notation, 4^2 means: “the number 4 multiplies itself”.

Now, if we do the same with the third piece, we will make 4 x 4 x 4 = 64 different images. Following the way of noting this, we have four multiplied by itself three times, or 4^3 = 4 x 4 x 4 = 64.

Now that you see the pattern, the big question is, when you multiply 4 times yourself, 300 times, what number do you end up with? Well, to show that, we need to introduce another form of exponential number – “power of 10”. This is probably familiar to you, because 10^2 = 10 x 10 = 100 = the number 1 followed by 2 zeros (the 2 is called the “pointer”). Similarly, 10^3 = 10 x 10 x 10 = 1000 = 1 followed by three zeros – so for exponents of 10, the calculator simply tells us how many zeros to put after the 1, so to enter the number. Every time a ratio increases by one, the number becomes ten times larger.

So, back to our original question: how big is a number 4^300? Well, it turns out that 4^300 is equal to this number: 10^180 – or the number 1 followed by 180 zeros! How big is that number? Really BIG! It’s so big, it’s bigger than the number of protons in the entire known universe. If you’re wondering about that number, it’s roughly 1.575 x 10^79. This is known as the Eddington Number. Follow that link to learn more about that, and other great numbers.

But, back to our puzzle. We now see that for one arrangement of pieces, simply rotating all the pieces to their four different positions – without changing their position, gives us the chance to create 10^180 different images… but we’ve only just begun! Take a few photos to know when you start moving the pieces, and to see a video demonstration of the Mona Lisa changing to Beethoven, visit the website link in the Resource Box.

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