Law of conservation of parity

The puzzle in the previous article involves arranging the pieces appropriately to form a target image.

Having confirmed that the basic operations worked, we set a goal: to arrange the components in a way that creates an aesthetically pleasing layout. We then chose the simplest possible initial configuration, but when we tried to solve the puzzle, we found that we couldn’t reach the intended goal.

Upon investigation, we discovered that the current initial configuration can never lead to the correct solution. The movement of the puzzle’s components is governed by an inherent parity (even versus odd positions), and this parity remains unchanged regardless of how the components are manipulated. Consequently, if the starting position corresponds to an “even” state and the goal to an “odd” state, the puzzle is fundamentally unsolvable.

We considered a mechanism to swap the parity, but since that didn’t seem very elegant, we instead worked backward from the goal state to determine a new initial configuration.