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It's pretty flat in terms of total elevation (highest point in Norfolk being Beacon Hill a mighty 105m above sea level) but as with most roads in the UK, there are deceptive undulations in a lot of places, and you might stumble across a short sharp climb or two along the coast.
I've ridden fixed from Norwich out to the coast too, and it's perfectly doable, but I'd say it's much faster/more enjoyable with gears. There's often a fair bit of wind in Norfolk, and the road surfaces aren't the best in places, so you can often find yourself grinding away for long periods.
Also, King's Lynn is a good 50 miles away from the Broads 'proper', so it'd be a reasonably big day to travel there and back fixed. Unless you're travelling between them in a car?
If you're just chilling out and want to do a couple of short/mid length rides, I'd go with the fixed if you're enjoying it. But if you want to go further afield, then maybe take gears.
Right this may be the place to ask rather than a fresh thread: off to the Norfolk broads, around King's Lynn, for a week starting tomorrow. Debating whether I should take the road bike or the fixed, which I have been fucking loving riding at the moment. I hear it's very flat so leaning towards the fixed, does anyone know of good routes out that way?