Leave this place and speak with the disgustingly ugly lady in front of the
Potion Shop wearing the blue dress, if she won't shut up about her dog then
you have the right person. Head down the alley on the opposite side as the
entrance you came in from to reach the Hyrule Castle area. Time proceeds
normally here so your only goal is to wait until it is night time. Enter the
Market again to find that most people are gone and there are a bunch of dogs
running around. Your goal is to find the fat lady's dog. It may be
randomized but for me, her dog was the white dog just left of the main
entrance to town, left when you are looking directly at the entrance. It was
running around just below that person in the window. Once you find the dog
you have to head down into the Back Alley. The Back Alley is located just to
the right when you're facing the main entrance. Enter the first door with the
dog following you (to get the dog to follow you simply walk up to it) and
inside this house is the lady. Assuming you brought her the correct dog she
will reward you with your first PIECE OF HEART [001]. Just like in the other
Zelda games, four Pieces of Heart will make up one Heart Container and
increase your maximum number of hearts by one. Be sure to note that you have
acquired this one on the checklist and all other ones you find in the future.