You’ll trigger this quest on the third day of Summer. Check your mail on that day and you’ll find a letter from Mayor Lewis, begging you to find his missing, purple shorts. There are no other clues besides the fact that they’re purple. 

The Lucky Purple Shorts are fairly easy to locate: they’re in Marnie’s room, at her Ranch. (And why might they be in there, hm?) You’ll find the Ranch one screen south of your farm. Problem is, you need to be friends with Marnie before you can enter her room, which means getting up to two hearts’ worth of affection. You’ll either need to present her with gifts or complete quests to do this, and since quests don’t pop up predictably enough gifts are the smart way to go. Marnie likes the following items:
  • Food items, particularly Cakes, Corn, and most cooked items
  • Valuable gems, like Diamonds
  • Eggs gleaned from a Coop
  • Milk gleaned from a Barn
  • Flowers
So long as you give Marnie ‘nice’ stuff and avoid obvious trash items you’ll probably appease her. Mind that you can only give someone two gifts per week, so it's better to wait and give her a higher quality item than nickel-and-diming her with lesser stuff.

Run into Marnie's room once you get up to two hearts, grab the Shorts, and return them to Lewis. He’s usually hanging out at his house, the Mayor’s Manor. Returning the Shorts will earn you a 750 gold reward via your Journal, as well as a heart worth of affection with Lewis.