Friday, December 31, 2010

The Brain Teaser: Assembly hint

This is part 5. After the fold I reveal one hint -- the blind spot that initially prevented me from assembling the puzzle given the manufacturer's solution.


You start assembling the puzzle using these two pieces. The piece in the upper left is labeled "A" in the manufacturer's solution, and there are two of them. The piece in the lower right is labeled "B" and there is just one.

The two pieces can be assembled into a cross by lifting A and placing it on B such that the two notches line up:


These two pieces now form an X-Y plane, where one is the X axis and the other is the Y axis. Take a moment to think about how many ways there are to do this. Notice that there are some configurations that violate the symmetry that will be required to solve the puzzle. In the above picture, look at the piece that is aligned left to right. It has one brown block running the full length, and two attached white blocks that are both pointing down. You could lift that piece up, rotate it 180 degrees and drop it back down into the notch, with the white pieces pointing up. But that configuration is lopsided so you can dismiss it as a possible solution.

The next step is to add the remaining "A" piece as the Z axis, like this:


Now, here is the hint. How many different ways can you assemble these three pieces together? Note as above that some of the arrangements are clearly illegal because they violate symmetry, i.e. they are lopsided in some way. Also, some of the arrangements are just rotations of other arrangements. If you ignore differences due to rotations, how many unique arrangements are there?

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