Resher: Thanks for the analysis.

Of course, there's no reason to believe this phenomenon is limited to opening rolls. In general, it seems one should bear in mind the enhanced probability of duplicate or similar rolls in planning strategy.

Has anyone run a test on the distribution of single die rolls? One way that these observed deviations from the norm could arise would be if, say, the chance of rolling a 4 on a single die was significantly higher than it should be. Depending on the pseudo RNG that is used, this might be a simpler explanation than any theory involving pairs of dice.