Label the coins
A1 B1 C1
A2 B2 C2
Here is the tree of weighings:
Weighing 1:
A1 B1 C1 | A2 B2 C2
Left lighter:
A1 | B1
Left lighter: A1 is fake
Right lighter: B1 is fake
Same weight : C1 is fake
Right lighter:
A2 | B2
Left lighter: A2 is fake
Right lighter: B2 is fake
Same weight : C2 is fake
Same weight: Impossible, since 5 of the 6 coins weighed in the first
weighing weigh the same, and the sixth is lighter.