RGG FAQ Part 5 Section 1
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Unfortunately, the method used for counting strongly depends on the ruleset used. (See faq section 1.2.) However, once they are decided upon, counting is only a matter of ordinary arithmetic. It is possible to rearrange territories in easily counted shapes (5 x n rectangles, for instance), but since this implies moving around the stones on the Goban, you should find out from your Goclub how and if this is done.
On the internet, servers (and/or clients) usually count the game for you semiautomatically. However, problems related to status of groups (life/death/seki ?) must be decided by the players. This is probably the weakest point of those servers w.r.t. possible cheaters (who might remove your live groups or leave their dead groups on the Goban). If you feel that something wrong is happening, you should ask a stronger player for help.
By the way, fine points of Japanesestyle rules are much concerned with counting of very complicated (and exceptional) positions.
For beginners, it is therefore often very much easier to learn the socalled 'Chinese' counting.
In case of lingering doubts, rec.games.go is the place to post.