Hello and welcome the the forum,
yerfedge! I think your variations starting with 1.Rg2 instead of the original solution of 1.Rh8 works out a-ok for Mate in 3 moves, good job. Just one thing though about your comment on the original solution-
Originally Posted by yerfedge
The problem with the two rook sacrifice is that the bishop can always opt out on taking the second rook. It can stay in the b8-h2 diagonal and the 3-move mate is in jeopardy.
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If the Bishop does take the first rook(and if it doesn't take the first Rook, it will still be Mate in 3 moves or less), then the Bishop will
have to take the second Rook as Black has no other move-
