Objectives
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Leave the lake raft where King Cormorant is seething, and swim (or utilize a boat to take you) back to shore. Gallop or run up the garden pathways, all the way to where the Duchess is explaining to her subjects that the rules of the game are being broken. Now you have all three clues, you can reveal a riddle that reveals the location of Milton. It reads thusly:
“I begin like a groan hollowed out with ease, then end like a mouse with the head of hard cheese.”
Speak to the Duchess, solving the riddle. “Groan hollowed out with ease” gives you “Green.” “Mouse with the head of cheese” could mean “house.” How about “Green-house”? This is the answer, and if you pick it, the Duchess is impressed with your powers of deduction. If you don’t pick it, she is less wowed by your lack of lateral thinking.
Either way, the Duchess knows of several greenhouses in the gardens, but only one of them has a door that locks; and the key you found looks to open it! She races off to find the structure as this quest concludes.