Earlier quoted context omitted.
> required to use powerful and energy-hungry GPUs to solve a complex mathematical problem. No mathematical problem is getting "solved". It is much more akin to playing the lottery: Entirely random numbers are being tried out, by sending them through an energy-intensive algorithm (being energy-intensive is the algorithm's only purpose in proof-of-work), until what comes out is less than another number. That other numb…
This is a lot of words to say that PoW involves an inverse-hash problem being solved. Inverse-hash is a mathematical problem. Finding the nonce is solving it.
Imagine a page of a book, for each letter a-z, assign a number to it (a=1, b=2, .. , z=26). Compute the sum of all of the letters on the page but reset back to zero when 100 is reached (i.e. 97 + 5 = 2). The resulting number is a kind of signature for the page. Changing a letter would result in a different sum.
At this point, people usually understand the utility and irreversible nature of it but quickly realize that many pages could have the resulting signature. It is a fairly easy leap of faith for most people to accept that there could be much more sophisticated algorithms that 1) use a bigger number for the signature and 2) take position into account and 3) result in very different signatures with tiny changes in input. This is usually sufficient for anyone to understand what is meant by a hash.
Finally explain that the hash must have a certain pattern to it (like end in a zero for example). If the hash of our "page" does not have this pattern, add another small gibberish word to the end and try it again. Keep trying until a hash with the expected pattern is found (and there is your Bitcoin).
The suggestion that a mathematical problem is being solved (while not completely inaccurate) sounds a bit more elegant in my opinion than what is really going on. It is really more like guessing (not that it really matters).