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Cuttack Badambadi (its local name) bus stand is about 6-8 km away from railway station. If don't have much of luggage, then take the sharing auto/tempo. Costs Rs 15-20. It varies as per auto to auto. And if you want to book, then 80-100 would be a fair deal, taking in consideration the ridiculous traffic of Cuttack city.
Biggest and Best government hospital of odisha, SCB medical college and hospital is in cuttack. And is hardly 3-4 km away. Take a sharing auto. It will cost Rs10. Or else if booking, then Rs 40-60 would be a fair deal. If in own vehicle, then the route will be via ranihat,manglabag. Turn right near ranihat square, you will reach back gate of hospital....
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If turned right from manglabag square, then you will reach main front entrance.
Just outside of railway station, maximum auto walas would be shouting two destinations-Badambadi and hospital.
Ravenshaw university (oldest university of odisha) is hardly 500m away from station. Come outside of the railway station and take the straight road ahead. If in doubt, ask the shopkeepers. Don't ask the auto walas as they will understand you are a stranger.