Five cousins are camping together. Two are from Montana, and three are from Wyoming. Two are Pattersons, and three are Andersons. Irene and Yvonne are from the same state, while Michael and Emily are from different states. Leo and Michael have the same last name; however, Emily and Yvonne do not. The Patterson from Montana awakes in the night to go to the outhouse. What is that cousin's first name?
Irene and Yvonne are from the same state
They cannot both be from Montana, because then the other three cousins would all be from Wyoming and
Michael and Emily are from different states
would be impossible. Therefore Irene and Yvonne are both from Wyoming. The same reasoning over the next two clues gets us that Leo and Michael are Andersons.
The Patterson from Montana
The Patterson from Montana is not from Wyoming so we can exclude Irene and Yvonne, and not an Anderson so we can exclude Leo and Michael. As there are only 5 cousins and we have excluded 4, we are only left with Emily.
Finally, we can show that the clues are consistent with an example.
Name: Emily Irene Leo Michael Yvonne Surname: Patterson Patterson Anderson Anderson Anderson State: Montana Wyoming Montana Wyoming Wyoming
And indeed this example fits all of the clues.
Their home states are either Montana or Wyoming, so Irene, Yvonne and one of Michael or Emily must be from Wyoming. Leo is from Montana.
Likewise, Leo, Michael and one of Emily/Yvonne are Andersons, and Irene is a Patterson.
Leo and one of Michael/Emily are from Montana, while Irene and one of Emily/Yvonne are Pattersons, with the common name being Emily.
Leo Anderson - Montana, Michael Anderson - Wyoming, Yvonne Anderson - Wyoming, Emily Patterson - Montana, Irene Patterson - Wyoming