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## Jigazo Puzzle – 300 Pieces Make Billions of Faces

The Jigazo puzzle – something new out of Japan – is a jigsaw puzzle consisting of a rectangular arrangement of 300 pieces, all the same size, 15 pieces wide, and 20 pieces high. Each piece has a single color, in varying degrees of intensity and gradation. Fragments are marked with unique icons. These icons allow pieces to be identified individually, so that they can be placed in the correct position to form an image by following the image map for the desired image. By arranging these pieces in the right way, virtually any image can be recreated.

In Japan, the word jigazo means “self portrait”. To create a self-portrait (or any other picture) with a Jigazo puzzle, simply email a copy of your picture (or any other picture) to the puzzle maker, and in a few minutes, you’ll receive a map. This map shows where each of the 300 pieces must be placed, and the proper orientation of each piece, to form the complete image. Of course, there is a limit to the amount of detail that the Jigazo puzzle can reproduce – but the fact that it works is incredible!

Okay, so now we’ve identified how a set of pieces of the same size but different color shades can be shuffled around to create different pictures – but how on earth is it possible that only 300 pieces can make a picture of someone? After all, there are about 7,000,000,000 people on Earth – surely one puzzle could not produce so many different pictures… could it?

Yes, it can happen – without even trying! In fact the number of different images in this puzzle can shock the imagination. The total number is so large that it exceeds any actual matching number in the known universe!

Let’s take a look at how this is possible: Start with an arbitrary arrangement of 300 pieces in the puzzle. That’s picture number one. Now, since all the pieces are the same size, each of those 300 pieces can be rotated 90 degrees and placed in four different positions. By doing so with the piece in the upper left corner, we will create four (never slightly) different pictures.

Now, in each of those four versions of the picture, we can take another piece in the top row, and rotate it to four different positions as well. This means that each of the four (very slightly) different pictures we created by rotating the first piece also has four different versions.

Now, you can see a pattern forming. Turning the first piece, we have 4 different pictures. For each of those 4 pictures rolling the second piece makes 4 pictures as well. So, for the first 2 pieces, the total number of pictures is given by 4 x 4 = 16. This can also be written as an exponential formula: 4^2 = 4 x 4 = 16. In this notation, 4^2 means: “number 4 multiplied by itself”.

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

Now that you see the pattern, the big question is, what number do you get when you multiply 4 times itself, 300 times? Well, to show that, we have to introduce another form of exponential number – “powers 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 “exponential”). Likewise, 10^3 = 10 x 10 x 10 = 1000 = three zeros after 1 – so for an exponent of 10, the exponent tells how many zeros to write after the 1 to write the number. Each time the exponent goes up by one, the number becomes ten times larger.

So, back to our original question: How big is 4^300? Well, it turns out that 4^300 equals this number: 10^180 – or 180 zeros after the number 1! How big is that number? Really great! It’s so big, it’s bigger than the number of protons in the entire known universe. If you’re curious about that number, its about 1.575 x 10^79. This is called the Eddington number. Follow that link to learn more about it, 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 positions, gives us the ability to create 10^180 different pictures…but we’ve only just begun! To find out how many pictures the puzzle can create as you start moving the pieces around, and to see a video demonstration of the Mona Lisa turning into Beethoven, visit the website link in the resource box.

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