Amit and Ian paint a wall in alternating shifts. First Amit paints alone, then Ian paints alone, then Amit paints alone, etc. During each of his shifts, Amit paints 1/2 of the remaining unpainted area of the wall, while Ian paints 1/3 of the remaining unpainted area of the wall during each of his shifts. If Amit goes first, what fraction of the wall's area will remain unpainted after Amit has completed his 6th shift? Choose one.
A) 1 / 27  B) 1 / 54  C) 1 / 81  D) 1 / 162  E) 1 / 486

Rationale: fraction of area unpainted after first shift of amit = 1 - ( 1 / 2 ) = 1 / 2 fraction of area unpainted after first shift of ian = ( 1 / 2 ) - ( 1 / 3 ) ( 1 / 2 ) = ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after second shift of amit = ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after second shift of ian = ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after third shift of amit = ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after third shift of ian = ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after forth shift of amit = ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after forth shift of ian = ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after fifth shift of amit = ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after fifth shift of ian = ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) fraction of area unpainted after sixth shift of amit = ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) ( 2 / 3 ) ( 1 / 2 ) = 486 answer : option e
Answer: E