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Article by Duncan Richer# The Eternity Puzzle

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Age 16 to 18

Published 1999 Revised 2011

The Eternity Puzzle is a new type of jigsaw. Unlike normal jigsaw puzzles, there is no picture - every piece is the same shade of green on both sides. All we know about the finished result is that it forms a regular dodecagon (12 sided polygon). The pieces don't have bumps or indentations, either, and all the edges are straight lines. This means that, when anyone looks at the puzzle, they can see that there many different ways to put two pieces together. An additional problem is that almost any two pieces can be placed together while leaving space for other pieces to go around them.

You can tell that this particular jigsaw has been designed to be extremely hard to solve. So hard, in fact, that the inventor has offered one million pounds to the first person who solves it, as long as they do it within the next five years. That is an awful lot of money just for completing a jigsaw, and you might think that it wouldn't take all that long.

However, when you first start trying to solve it, you'll soon see that there are far too many ways to start which go wrong. Firstly, though, a couple of things to point out about how you can improve your chances of going right.

With the puzzle you get a backing sheet of paper with some grid lines on it, as well as the exact location of one of the 209 numbered pieces. All pieces differ in shape, so being able to put a unique piece in position will help at least a little. There are also three much smaller puzzles available to buy, similar in idea but with far fewer pieces (less than 30 pieces each). If you solve those then you are told the locations of additional pieces in the Eternity puzzle, so you can fix 4 or 5 of the piece positions immediately.

The grid lines on the backing paper are also very useful. The backing paper is drawn up into equilateral triangles, just like isometric paper with three sets of parallel lines drawn on it. Each of the pieces can be placed on this grid so that the edges either go along the grid lines, or cut the equilateral triangles exactly in half. So every piece can be oriented in 12 different ways, only one of which will be right.

The number of ways to orient these pieces, even if you get all the clues available, is 12 ^{204} . That's just trying to get all the pieces placed at the correct angles, not even trying to put them together on the board! When we start trying to put pieces together, the number of different ways to try becomes truly staggering!

It is extremely hard to come up with an exact number of ways of putting the pieces together "wrongly". To count them we would need to go through exhaustively checking each case, adding pieces until we couldn't add any more correctly, then taking out one of the pieces and trying again. The estimate I came up with for the total number of ways to attempt to solve it was 10 ^{500} . So if
you tried, just once, to solve the Eternity puzzle, then your odds of getting the million pounds would be about 1 in 10 ^{500} . Compare this to the odds of the National Lottery - 1 in 14 million. The odds of getting this puzzle right, first time, are about the same as the odds of the same set of 6 numbers coming up as the National Lottery numbers every Saturday for a year and a half.

Those are just the odds if you try it once. So you might think you could just get a computer to try all the options, and it won't take very long to find the right one. It's a nice idea, and in many problems it's the right way to go. However, the number of different ways to attempt Eternity is so large that even having hundreds of thousands of computers helping out won't really do you much
good. If you had one million computers, each testing out 50 million possible ways to solve the puzzle every second, then every day you would be testing less than 10 ^{19} possibilities. At that rate, it would take the computers longer than the age of the universe to sort through all the possible solutions.

As far as I can tell, the million pounds looks safe.

It turns out that Duncan was wrong, and that it was a pair of mathematicians that solved the puzzle. For the Plus article about how they did it, click here.

For more about this puzzle and the solution to it see the http://www.mathpuzzle.com/eternity.html site.

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