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 Well, things are heating up. I'm afraid I cannot agree with Paul at all
 and will outline some reasons. Kevin Padian, not unexpectedly, makes good
 sense and some continued high-level functional analyses combined with
 more phylogenetic and historical/phylogenetic analyses will help tease
 the answer out, especially as more material, including the new near
 complete Pachycephalosaurus with the horns and post-cranial material
 but  lacking the horn due to erosion will help as it get described.

 Paul was unsure of the arguments for sexual dimorphism in pachys. Well,
 Pachys have the best developed and described sexual dimorphism in the
 whole of the Dinosauria (excluding the avians). Dodson's work on the
 Protoceratops and lambeosaurines was excellent but the former is not
 not as convincing (but still convincing to me) and the latter are tough
 to relate specific sexes with each other. Stegoceras, on the other hand,
 has about as perfect case as one can find - and I'm not saying this with
 lack of modesty because I described it, Stegoceras did all the work.
 The reference:

 Chapman, Ralph E., P.M. Galton, J.J. Sepkoski, Jr. & W.P. Wall. 1981.
    A morphometric study of the cranium of the pachycephalosaurid
    dinosaur Stegoceras. J. Paleontology, 55(3):608-618.

 Also, the data showing that pachys were impacting greatly through their
 domes is solidly settled by the work of Galton, Wall, Hans Sues, Maryanska
 Osmolska, etc. The discussion then settles away from brush running to
 either defensive or sexual combat functions. The former is an option
 regardless of the latter but would only be desperation, in my opinion.
 The best D against a large theropod for most pachys is to get away from it
 as best as possible. Charging towards it would be suicidal in most
 cases. That leaves sexual dimorphism as the primary shaper, bolstered by
 the bumps and horns associated with many taxa and the new Pachycephalosaurus.
 For up close head butting, glancing blows are immaterial. The whole
 combat would include those, side blows, locking of bumps and horns, and
 straight on head-bashing. Flank attacks are also reasonable, maybe more so,
 but I see no strong argument yet against head/head given close-up
 front combat. Running a 1/4 mile before bashing would be another story and
 I agree that seems out of the reasonable area. We just need to develop
 strong functional and phylogenetic models to determine, if we can, whether
 either or both are favored among the flank/head-on models. I agree with
 Kevin in that I think we can get there.

 Yours truly,

 Ralph Chapman, NMNH