Solving problems is certainly valuable.
Coding, at a senior level, is worthless, IMHO.
I hate hand-writing. I can't do it for longer than maybe 10 minutes, without getting annoyed. There are no legitimate problems that I solve daily that can be written in code on paper. Either the code is so obvious that it's a waste of time to write it (i.e. Serialise this thing, write this collection). The things I write down in day-to-day work are plans and ideas, not actual code.
I can be fairly arrogant, and a few years ago I was asked to solve some silly puzzles (write a tic-tac-toe game) and I effectively walked out and told them just how silly I thought it was. So for me, asking me to write code is an effective way to have me leave. If that's what you're after, good for you :)
On the other hand, I have been asked to write code at a computer; and I'm more than happy to do that. I've also been asked to solve general problems on a whiteboard, and I'm happy with that as well.