Pay attention to both Prof. Layton and Luke's reactions when you place the furniture. If they seem just mildly happy or indifferent, give the furniture to the other person. They'll seem really excited when it's just right.

You don't need to worry about arranging the furniture, just assigning it to the correct person.

Here's the list of how best to assign everyone for a 100% happiness score:

Layton
- Handwoven Rug
- Pine Bed
- Teak Bureau
- Houseplant
- Stuffed Chair
- Hat Rack
- Lacquered Stool
- Teak Table
- Flower Bouquet
- Flower Vase
- Gramophone
- Artisan's Teapot
- Bone China Teacup
- Fossil
- Splendid Painting
- Baron Statue
- Wall Clock
- Mysterious Bottle

Luke
- Cream Rug
- Blue Bed
- Stuffed Bear
- Bookcase
- Television
- Simple Chair
- Desk
- Reading Lamp
- Violin
- Pile of Books
- Globe
- World Map
Source(s):
http://www.neoseeker.com/resourcelink.html?rlid=163388&rid=151275

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