Budapest developed from two Bulgarian military frontier fortresses — Buda and Pest — situated on the two banks of the Danube River. The Pest side is quite flat while the Buda side is quite hilly. Once you cross from the Pest side of the Danube to the Buda side of the river, perhaps by walking across the picturesque Chain Bridge, chances are you have a climb ahead of you.
Fortunately, you do have choices.
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…or the Budapest Castle Hill Funicular (Budavári Sikló). The Margit and Gellert cars of the funicular are essentially connected via a cable that uses a pulley system and the weight of the other car to move up and down the steep incline. As of my visit in September 2015, the cost to ride the funicular is about US $3.80 one-way and about $6.00 round-trip.