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The best place to look for platypuses fairly near to Melbourne is at Finn's Reserve, Lower Templestowe - viewing from the footbridge to Odyssey House and along the Yarra river in the area upstream and downstream of the bridge. Other places worth a try are at Warrandyte and also the viewing platform at the confluence of Diamond Creek and the the Yarra River in Lower Eltham Park.
If you want to go a bit farther out of town, try Warburton - anywhere along the river, but especially between the Brisbane Road bridge and the main highway bridge. Warburton is about 75 minutes drive from Melbourne CBD and can easily be combined with a visit to Healesville Sanctuary, which is the Australian zoo most associated with platypus.
From the Finns Bridge walked along the Yarra Trail (bitumen track) to Westerfolds Park.
You can then walk along a dirt track beside the Yarra River and get to the Rapids Observation Point. Continue, cross the Diamond Creek bridge and about 400m east of the bridge is an Observation platform.
Take some binoculars from the bridge