One group started with this problem:
"What if the constructed cube had different colors on each of the six sides? Would that change the number of nets that are possible?"
"What if we had six different colors? How many 'different' cubes could we create by painting the faces with those six colors?
Another group started with this problem:
"What if our box didn't need a lid? Would that change the shape of the packing that uses the least amount of material?
"What if we only needed to pack half as many cubes (12)? Would we need half as much material to make the box?
There's lots to think about here, but there are a couple things that stand out to me:
1. Culture of Curiosity - We hear a lot about "open-ended problems", but it doesn't seem to me that problems themselves are ever actually open-ended. Rather, it's the person that makes a situation open-ended by changing things about the conditions, investigating variants and invariants, posing new and related questions, and, generally, just being curious. So, I think I'm advocating for an "open-ended culture" rather than thinking about problems as supplying that.
2. The Two Part Extension - I noticed that each of the extension questions that were posed followed a similar format that went something like:
"What if ___________ ? (How) would that change _________ ?