Well, as promised, here's the solution:
Reference time will be UTC. As c² said, the trick is starting off near the international date line, then going south (over the south pole), then returning to the date line (this time more to the north so that you get on the other side of it).
See:
http://upload.wikimedia.org/wikipedia/c ... e_Line.png
As you see there, Samoa and Tonga are both in the +13 UTC zone; Baker Island in the -12 zone - only 25 hours. So what is missing?

The fact those guys have summer down there at the moment. And, some of them, use summer time. Samoa does, Tonga does not

So Samoa is actually at +14 UTC. (
http://en.wikipedia.org/wiki/Summer_time)
That means: (Sa was 31.12.11, So = 1.1.12; dst = daylight saving time)
Sa 10:00 UTC - Samoa (dst), +14 UTC, 172°80'W
Sa 11:00 UTC - Tonga (no dst), +13 UTC, 178°55'W (Minerva Reef)
Sa 22:00 UTC - Sanae IV station (no dst), +2 UTC, sun touches horizon, 2°50'26''W
So 00:00 UTC - near south pole, +0 UTC, sun shining, 0°W
So 12:00 UTC - Baker Island (no dst), -12 UTC, 176°28'W
The longest travel is about 1/4 around the globe, that is 10000 km, speed of sound is 1236 km/h, so ~ 8 hours flight (from Tonga to Sanae IV for example) - seems to be enough time.
So if you have any logical complaints about this, tell me
c-square, it's your turn!