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The answer makes sense to me. However, what if $\ce{OsO4}$ were to attack from the other side of the double bond? Why does the answer key not show that possibility? Would we not get an other molecule?

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    $\begingroup$ Attack from either side produces the same molecule. $\endgroup$
    – canadianer
    Commented Sep 27, 2014 at 15:54

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If you look carefully you'll see that your compound has stereocenters, yet is achiral. Both stereocenters have the opposite configuration. This is the meso-form, these are achiral, and the hydroxy groups are prochiral. Look carefully for the rotoinversion center!

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