Achilles: Is 15 wondrous, or unwondrous? Let's see:15 IS ODD, so I make 3n + 1: 46
46 IS EVEN, so I take half: 23
23 IS ODD, so I make 3n + 1: 70
70 IS EVEN, so I take half: 35
35 IS ODD, so I make 3n + 1: 106
106 is EVEN, so I take half: 53
53 IS ODD, so I make 3n + 1: 160
160 IS EVEN, so I take half: 80
80 IS EVEN, so I take half: 40
40 IS EVEN, so I take half: 20
20 IS EVEN, so I take half: 10
10 is EVEN, so I take half: 5
5 IS ODD, so I make 3n + 1:^16
16 IS EVEN, so I take half: 8
8 IS EVEN, so I take half:^4
4 IS EVEN, so I take half: 2
2 IS EVEN, so I take half: 1.Wow! That's quite a roundabout journey, from 15 to 1. But I finally
reached it. That shows that 15 has the property of being wondrous. I
wonder what numbers are uNwondrous ...
Tortoise: Did you notice how the numbers swung up and down, in this
simply defined process?
Achilles: Yes. I was particularly surprised, after thirteen turns, to find
myself at 16, only one greater than 15, the number I started with. In
one sense, I was almost back where I started-yet in another sense, I
was nowhere near where I had started. Also, I found it quite curious
that I had to go as high as 160 to resolve the question. I wonder how
come.
Tortoise: Yes, there is an infinite "sky" into which you can sail, and it is very
hard to know in advance how high into the sky you will wind up sailing.
Indeed, it is quite plausible that you might just sail up and up and up,
and never come down.
Achilles: Really? I guess that is conceivable-but what a weird coincidence
it would require! You'd just have to hit odd number after odd number,
with only a few evens mixed in. I doubt if that would ever happen-but
I just don't know for sure.
Tortoise: Why don't you try starting with 27? Mind you, I don't promise
anything. But sometime, just try it, for your amusement. And I'd
advise you to bring along a rather large sheet of paper.
Achilles: Hmm ... Sounds interesting. You know, it still makes me feel
funny to associate the wondrousness (or unwondrousness) with the
starting number, when it is so obviously a property of the entire
number system.
Tortoise: I understand what you mean, but it's not that different from
saying "29 is prime", or "gold is valuable"-both statements attribute toAria with Diverse Variations 401