Collatz’ conjecture

 Number crunchers know that the word conjecture is always a warning that by the pricking of my sums, something evil this way comes. Conjectures are unsolved problems, and in fact, Paul Erdös, a noted Hungarian mathematician, was reported to have said of Collatz’ conjecture, “Mathematics may not be ready for such problems.” Others called it “dangerous” and “a quagmire”.

When it comes to mathematical challenges, the Four-colour map problem, Fermat’s last theorem and squaring the circle, are far too difficult to even consider on a bus, but the Collatz conjecture is nice and simple to play with. It was put forward by Lothar Collatz, who waited two years after receiving his doctorate, before offering this puzzle. Pro tip: always get your higher degree nailed to the wall before you make waves!

Choose any positive integer n to begin a series. For each following term, if the previous term is even, the next term is one half of the previous term. On the other hand, if the term is odd, multiply it by 3 and add 1. Collatz’ conjecture is that no matter what the value of n, the sequence will always reach 1. Here are five sample strings:

1, 4, 2, 1;

2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1;

3, 10, 5, 16, 8, 4, 2, 1;

4, 2, 1;

5, 16, 8, 4, 2, 1.

The sequences generated are sometimes called the hailstone sequence or hailstone numbers, because the values usually go through multiple ascents and descents, like hailstones in a cloud.

If you are working through the numbers on your bus ride, can you see what the next number is that you need to test? From what you can see above, you can rule out 7, 8, 10, 11, 13, 16, 17 and lots more…

The Hungarian-born mathematician Paul Erdös (1913–1996), is considered to hold the world record for the number of papers he wrote in collaboration with other mathematicians. Erdös numbers are whimsical numbers given to mathematicians. Erdös himself has the Erdös number 0, and any person who has collaborated with Erdös on a paper has an Erdös number of 1, while a mathematician who has collaborated with a direct collaborator is given an Erdös number of 2, and so on.