Collatz sequences

Look at this sequence: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. Can you see the rule? Start with any whole number:

• if it is even, take half of it;
• if it is odd, multiply by 3 and add 1.

Then do the same with the new number, again and again.

Check it: 12 is even → 6. 6 is even → 3. 3 is odd → 3 × 3 + 1 = 10. 10 → 5. 5 is odd → 5 × 3 + 1 = 16. Then 8, 4, 2, 1. Another of the book's sequences: 21, 64, 32, 16, 8, 4, 2, 1.

Start with 16, 32, 64 or any number in the doubling sequence 1, 2, 4, 8, 16, …, and you only ever halve, all the way down to 1.

In 1937 the German mathematician Lothar Collatz guessed that the sequence always reaches 1, whatever whole number you start with. This guess is called Collatz's conjecture. Every number tried so far does reach 1. But even today, nobody knows whether it is true for all numbers: it is one of the most famous unsolved problems in mathematics. Maybe one day you will solve it!