We are often overscheduled and obsessed with efficiency. A plethora of opportunities awaits and we are eager to seize them all, as though we fear if we allow ourselves to squander a few minutes, the world is at risk of changing without us.

Where do ideas come from? That's a big question. Here's a smaller one: Where do mathematical ideas come from? I've wondered about this from the time I first contemplated being a mathematician until long after I officially became one. My earliest memory of anything like a mathematical idea comes from a childhood walk with my dad.

Henri Poincaré, the father of chaos theory and the co-discoverer of special relativity, relates his own discovery. "the idea came to me, without anything in my former thoughts seeming to have paved the way for it.”

The Irish mathematician, Sir William Rowan Hamilton, had a similar epiphany while strolling by the Brougham Bridge. He was so delighted that he stopped and carved the defining algebraic equation into the bridge.

The first phase of solving can be described as “worrying” about a problem or idea. It evokes anxiety and gives the impression of productivity.

But, overthinking can lead to a dead end. The key to solving the problem is to take a break from worrying. Focus your attention on some other activity. Take a long hike or a long drive, to give your mind the space to have a good idea.