The 3x + 1 problem: Here is a mathematical function f (n)that applies only to whole numbers n.
If a number is even, divide it by 2. If it is odd, triple it and add 1. For example, 16 is even, so we divide by 2: f (16) = 16 2 = 8. On the other hand, 15 is odd, so we triple it and add 1: f (15) = 3 × 15 + 1 = 46.
a. Apply the function f repeatedly beginning with n = 1. That is, calculate f (1), f (the answer from the first part), f (the answer from the second part), and so on. What pattern do you see?
b. Apply the function f repeatedly beginning with n = 5. How many steps does it take to get to 1?
c. Apply the function f repeatedly beginning with n = 7. How many steps does it take to get to 1?
d. Try several other numbers of your own choosing. Does the process always take you back to 1? (Note:We can’t be sure what your answer will be here. Every number that anyone has tried so far leads eventually back to 1, and it is conjectured that this happens no matter what number you start with. This is known to mathematicians as the 3x + 1 conjecture, and it is, as of the writing of this book, an unsolved problem. If you can find a starting number that does not lead back to 1, or if you can somehow show that the path always leads back to 1, you will have solved a problem that has eluded mathematicians for a number of years. Good hunting!)