For those who don't understand the above sentence, (0,1) is the set of all numbers in-between 0 and 1, while [0,1] is the set of all numbers between 0 and 1, including 0 and 1 themselves. You must find a function that matches every number in the first set with a single number in the second set so that each number in each set is used exactly once.

Map [0,1] to itself, except for 0, 1 and positive integral powers of 1/2 and 1/3.

Map 0 to 1/2 and 1 to 1/3.

Map all (1/2)^n to (1/2)^(n+1) for all positive integral n.

Map all (1/3)^n to (1/3)^(n+1) for all positive integral n.