Start with any number.
If your number is even, cut it in half.
If your number is odd, triple it and add 1.
Write down your new number, and repeat.
Do you always end up at 1, 4, 2, 1, ... ?
Collatz's Conjecture was that you do. No one has found a counter-example in the century or so since he proposed his conjecture. No one has a proof that it will always happen either.
Kids like playing around with this. So I like it. But I hadn't played around much with it until now.
Discrete Mathematics With Ducks mentions this problem / puzzle / conjecture in Chapter 5, on algorithms. So I made a spreadsheet, and am looking at the lengths of the series for each number up to 100. I noticed two things:
- Quite a few consecutive pairs of numbers have the same length series. (I'm counting how long until the number 1 shows up.) It seems to always be an even number and then the number after it.
- I haven't counted them all yet (I'm doing it by hand), but there seems to be a big gap. Most sequences have lengths under 25, and a few have lengths over 100. So far, I haven't found any sequence lengths in between 25 and 100.
What do you notice about the Collatz series? What do you wonder?