okay, I'm just not seeing how a puzzle with pieces that can be placed on the other side meets that property. The pieces would have to be elastic, and if the pieces are allowed to change shape, it's not really a puzzle anymore.
That's why I wrote that you have to imagine the surface formed by making every possible connection simultaneously. The point is that the "completed" puzzle is topologically a Klein bottle. It can't be completely constructed in 3 dimensions.
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okay, I'm just not seeing how a puzzle with pieces that can be placed on the other side meets that property. The pieces would have to be elastic, and if the pieces are allowed to change shape, it's not really a puzzle anymore.
That's why I wrote that you have to imagine the surface formed by making every possible connection simultaneously. The point is that the "completed" puzzle is topologically a Klein bottle. It can't be completely constructed in 3 dimensions.