There are some places (e.g. India) that are half an hour off of the time zones around them. Why is this?
The Perfect Master speaks.
The South Australia/Northern Territory time zone is also a “half time” zone, at GMT + 9.5 hours. In summer, South Australia has daylight saving and goes to GMT + 10.5 hours.
Expanding a bit on this …
Australia has three time zones, … in winter. In summer it has five time zones! Not only is it possible to travel east and west across the country and the time will change, but one can also travel north and south and the time will change as well. And the last time I check, not all areas of Australia begin and end daylight saving at the same time, either!
The reason for this is that there are eight state and territory jurisdictions in Australia, each of which can decide whether or not to observe daylight saving. New South Wales (NSW), the Australian Capital Territory (ACT), Victoria (VIC), Tasmania (TAS) and South Australia (SA) all have daylight saving. Queensland (QLD), the Northern Territory (NT) and Western Australia (WA) do not have daylight saving.
This leaves the summer time zones as:
GMT+11 (NSW, ACT, VIC, TAS)
GMT+10.5 (SA)
GMT+10 (QLD)
GMT+9.5 (NT)
GMT+8 (WA)
From the above you can see that if you go north across the NSW/QLD border, you lose an hour, and if you go west across the NSW/SA border, you lose half an hour. Even better, if you go west across the QLD/SA border, you gain half an hour despite the fact you’re going west.
TAS is further south. It benefits more from daylight saving and so starts earlier, on the first Sunday in October. The other states and territories wait until the last Sunday in October to begin daylight saving. All states and territories end daylight saving on the last Sunday in March.
Newfoundland is also a half-hour step away from Atlantic time. Since the CBC staggers its newscasts so every time zone gets the “News at 11”, the addendum always has to be made, “one-half hour later in Newfoundland”. This has led to a popular joke: “Scientists have just discovered that the world will end when a comet smashes into the earth tonight at midnight, 12:30 in Newfoundland” 
If that isn’t weird enough look up Chatham Island at one of my favorite sites:
www.timeanddate.com
It runs 45 minutes ‘ahead’ (or 15 minutes 'behind") the standard time zones. I believe Chatham Island is East of New Zealand and is sparsely populated. Anyone know why it has this unique horological quirk?
I’ve been there and yes, the Chathams are sparsely populated. They’re 800km east of New Zealand, so I suppose they’re far enough east for the time difference to be formally recognised via a separate time zone.