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I am struggling to find a mechanism for the above reaction.

So far I have suggested that the proton adds to one of the reacting aromatic species leaving the resonance stabilized secondary cation. This then adds to the ortho position on the second reactant (since alkyl groups are activating). I then get a positively charged intermediate.

For a simpler reaction I would simply remove the proton but I wonder whether the reaction from this step to the product is concerted? I couldn't see an obvious step so I deprotonated and again could find no obvious way to continue apart from a rather elaborate proton transfer from a carbon that does not possess the most acidic proton.

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You're right that the first step involves protonation of the terminal carbon in the styrene double to yield a resonance stabilized benzyl carbocation. This carbocation then adds to another styrene molecule in the exact same fashion, reacting at the terminus of the styrene double bond to again yield a resonance stabilized benzyl carbocation. This latter carbocation then alkylates itself (like a Friedel-Crafts alkylation). Loss of a proton from this final carbocation yields the indicated product.

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    $\begingroup$ The mechanism depicted above is most likely correct but please do not push arrows this way. All the arrows are backwards and suggest electrons originating from carbocations which are most definitely sinks and not sources. $\endgroup$ – Simon Cooper Jan 11 '18 at 2:33

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