That little white castle-looking building near the water? That is Square Tower. Most people walk past it, snap two photos, then only later realise it has been sitting there since the Brooke era.
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Source: Wikimedia Commons - Square Tower 01
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Source: Wikimedia Commons - Square Tower 02
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Source: Wikimedia Commons - Square Tower 03
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Source: Wikimedia Commons - Square Tower DSC05966
Real talk first: do not plan this like a full museum visit. Square Tower is more of a 15-30 minute heritage stop while you are already around the old town area. The outside is the main thing: white walls, wooden shutters, Brooke-style details, and a nice angle with the State Legislative Assembly building across the water.
My take: worth stopping if you are already doing the Waterfront / Main Bazaar / Old Courthouse walk. Not worth driving across town just for this one building only, unless you really enjoy old architecture or want a clean photo spot.
Quick tips
Best time: around 5:00 PM onwards, less hot and better light.
Time needed: 15-30 minutes.
Ticket: free for the outside area.
Access: exterior is on the public walkway; indoor access can change.
Nearby: Old Courthouse, Main Bazaar, Darul Hana Bridge, Chinese History Museum.
Photo angle: try the side with the golden DUN building in the background.
If you have walked past this place many times but never stopped, maybe next time give it five minutes. Sometimes Kuching's small old buildings quietly carry more story than the big tourist spots.