  We prove three conjectures concerning the evaluation of determinants, which
are related to the counting of plane partitions and rhombus tilings. One of
them was posed by George Andrews in 1980, the other two were by Guoce Xin and
Christian Krattenthaler. Our proofs employ computer algebra methods, namely,
the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These
variations make Zeilberger's original approach even more powerful and allow for
addressing a wider variety of determinants. Finally, we present, as a challenge
problem, a conjecture about a closed-form evaluation of Andrews's determinant.
