### Alexandre Muñiz

Shared publicly -Here's a new type of puzzle I'm working on. Given five transparent 15-omino pieces, make a complete set of pentominoes by overlapping them. The solution for this particular set of pieces is shown below. There are a zillion ways to arrange the pentominoes into five and seven piece clusters that could produce a puzzle of this form. The real problem is, can we find a particularly good set of 15-ominoes to use?

What makes a particularly good set? As much as possible, the 15-ominoes should be partitionable into three different pentominoes in multiple ways. (These partitions must have a middle pentomino that leaves two other pentominoes when it is cut out.) Also, as many as possible of the pentominoes should be present in dissections of more than one 15-omino. It'd also be nice to have at least one false partial solution, like a pair of 15-ominoes that can make 5 different pentominoes by overlapping that isn't present in the correct solution.

Then there are some practical matters related to making this puzzle as a physical lasercut set. It would be bad if the centers of gravity of the pieces in the pair were not within the pentomino of overlap, because the top piece could fall over. I'd also want the 15-omino pieces to have an efficient packing into a square, in order to minimize wasted material and cut length.

What makes a particularly good set? As much as possible, the 15-ominoes should be partitionable into three different pentominoes in multiple ways. (These partitions must have a middle pentomino that leaves two other pentominoes when it is cut out.) Also, as many as possible of the pentominoes should be present in dissections of more than one 15-omino. It'd also be nice to have at least one false partial solution, like a pair of 15-ominoes that can make 5 different pentominoes by overlapping that isn't present in the correct solution.

Then there are some practical matters related to making this puzzle as a physical lasercut set. It would be bad if the centers of gravity of the pieces in the pair were not within the pentomino of overlap, because the top piece could fall over. I'd also want the 15-omino pieces to have an efficient packing into a square, in order to minimize wasted material and cut length.

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