I took up a small CSS challenge to solve for a client and I'm going to be paid on a hourly rate. I eventually solved it, it took 4 hours but I spent roughly 30% of the time in the wrong track, trying a CSS3 solution that only worked in recent browsers and finally discovering that no fallback is possible via JS (like I originally thought). Should I charge the client that 30%?

More details: I didn't provide an estimate, I liked the challenge per se, so I started working on it before giving an estimate (but I have worked with him before, so I know he's not one of those people that have unrealistic expectations). At the very worst I will have spent 4 unpaid hours on an intriguing CSS challenge. And I will give the fairest possible estimate for both of us, since I will have already done the work. :)