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During the dance, the new index is transferred from one server to another, with the effect of the data centres updating one by one. Because a search at google.com returns results from one of the data centres and during the dance their can be different data on different servers, it's possible to see the old and new results on Google between searches (this variation = dancing ;))
www2 and www3 are not servers - they are merely domain names set to show results from one or other of the data centres, presumably for testing purposes, and convenience.
Unfortunately there is no fixed pattern or order to when the different datacentres update with the new index, so the best bet is to use a tool to query all of the data centres and see which ones have new backlink counts.
And lets just remember that since Dominic [webmasterworld.com], all bets are off.