There are two types of thinking to use here: lateral and logical
To use lateral thinking, you need creativity and out of the box thinking and for logical thinking you will need to work out the problem linearly or use more analysis.
The best tool to use for finding the solutions is brainstorming:
Most people do their brainstorms wrong in one simple way, but it's really important. And that is that they judge each idea as it comes up, instead of just getting lots of ideas in Phase 1, and then completely separately in Phase 2, "judging the ideas". It's human nature to judge each idea, but you must resist the temptation. Just write each one on the list as if it's fine, even if it's utterly stupid. Get every idea on the list. And there are four reasons for doing it like this.
First, is that you can bounce from a stupid idea to a great one.
So who knows where that stupid idea might lead?
Second, is that it puts a damper on people's creativity if their ideas are criticized. And you want to keep them rolling, keep all those ideas coming.
Maybe they will have to pass through nine bad ideas before they come up with the good one. And you don't want to switch them off after the first nine, or perhaps after their first one by telling them that their ideas are not good enough.
The third is ownership. People waste time and bias the process by arguing for their own ideas. They naturally defend their own ideas.
But if you just make a great big list and then come back to judging it later, people don't feel the need to defend their own ideas so much. They often can't even remember whose idea was which. So they can properly judge each one, and choose the actual best one. You don't just end up with the one that was suggested by the most dominant person.
Finally, the fourth reason is that we're using different parts of our brains for the two phases.
The first, creative phase, is using what's known as the "right brain," although it's not really on the right of our heads. That's a myth. But that childish, fun, creative part needs to be on maximum. Then, in the second phase, we use the logical, adult, sensible, often-called "left brain" to judge the ideas. And this is a completely different mode. It will be really hard to swap between the two modes constantly. In fact, most people in a bad brainstorm just stick to one mode.
So a few people are suggesting ideas, and the others are judging them. So you lose the creativity of half the group, and the creative ones just start to hate the others. So, much better to have everyone being creative, and then later, everyone judging. So there we are. That's all we need to know about brainstorming. Everyone coming up with ideas, written up on a board with no judging, no criticizing allowed. And then later, you go through them and assess which ones are best.