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In Grindelwald, a short distance on the map can hide a lot of climbing. Choose a signed bike route by its ascent and surface, then hire a bike suited to it. The Eiger gives you a good reason to stop, so do not fill the day with a route at the limit of your fitness.
Technical trails call for the skills to ride them, and even a short route can occupy much of a day. Ask a rental shop about the bike and route together. If your plan includes a lift, bicycle carriage depends on the route and date.