What happening when Bilbo workout how to open the secret door?

In "The Hobbit," Bilbo Baggins discovers how to open the secret door to the Lonely Mountain after much contemplation and reflection on the clues provided by the map and his surroundings. He recalls the phrase "the last light of Durin's Day," realizing that the door can only be opened at sunset on that specific day when the last rays of light illuminate the keyhole. As he shares this revelation with Thorin and the other dwarves, they eagerly gather around, and Bilbo's insight ultimately leads them to the entrance of their ancestral home. This moment highlights Bilbo's growing resourcefulness and importance to the quest.