My nephew Matthew sent me a number riddle (thanks, Matt!), which I pass on here:
1 is 3, 3 is 5, 5 is 4, and 4 is cosmic.
Why is 4 cosmic? What happens to the other numbers?
As is the case for most riddles, the answer is all over the internet, so give this one a good try if you wish to solve it for yourself. I slept on it for several days, spending a few moments of thought on it each day, until the answer came to me in a flash. The feeling of satisfaction when the flash of insight comes is well worth the wait, so if you’re interested I encourage you to just play with it and then sleep on it if you don’t get it right away.
The greatest mathematicians and scientists report similar flashes of insights that helped them solve great problems and create great advances. My own thesis supervisor, himself a great man, told me that when you work hard enough for long enough you begin to solve problems in your sleep.
As educators, we need to find a way to encourage/train students to be patient with themselves, and to persist in working on problems for however long is necessary to solve them. This would be more beneficial to their intellectual development than any content knowledge we stuff into them.
I’ll post an answer and a few comments on related mathematics in a few days.