The guys who could calculate it electronically lost interest in this thread.

Here you go...

PROVNAME CentroidX CentroidY
Eastern Cape 26° 26' 46.898" E 32° 9' 14.499" S
Free State 26° 52' 2.150" E 28° 36' 51.852" S
Gauteng 28° 11' 51.183" E 26° 4' 31.572" S
KwaZulu-Natal 30° 44' 54.812" E 28° 42' 59.707" S
Mpumalanga 30° 17' 56.335" E 25° 52' 25.144" S
Northern Cape 21° 24' 10.114" E 29° 27' 12.258" S
Limpopo 29° 17' 23.148" E 23° 47' 7.207" S
North West 25° 32' 54.452" E 26° 16' 13.495" S
Western Cape 20° 36' 57.737" E 33° 0' 52.372" S
South Africa 25° 5' 38.087" E 28° 59' 43.146" S