While some problems are actually solvable with just pencil and paper, this is not a requirement and the intention of the project is that problems will be solved with a combination of insight and coding.

From the Project Euler website:

"Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems."

Well you probably _could_ whiteboard it but some of them are about extending a pattern to the millionth term. If you're really smart you might be able to find a closed form to get there quick, but you can also just program the basic rule and iterate

For the harder Euler problems you usually need a combination of mathematical insight and good coding technique. Each problem typically provides sample values for smaller instances of the problem, which you can confirm with a straightforward solution approach. Then the actual problem will be to calculate f(10^500) or some absurd value that requires insight to make tractable. They are extremely good challenges overall.