Wondrous Numbers

Date: 02/07/2001 at 20:50:39
From: Aaron Hass
Subject: What is a wondrous number

What is a wondrous number?

Date: 02/08/2001 at 17:06:08
From: Doctor Achilles
Subject: Re: What is a wondrous number

Hi Aaron,

There may be more than one definition of "wondrous number" out there, 
but the only one I've been able to find comes from Douglas 
Hofstadter's book _Godel, Escher, Bach_ (a personal favorite). I have 
the Basic Books 20th Anniversary paperback edition, which introduces 
wondrous numbers on pages 400-401.

Wondrous numbers are defined like this: Start with an arbitrary 
natural number (integer greater than zero). If the number is even, 
divide it by 2. If it is odd, multiply by 3 and add 1.  Repeat until 
you come out with 1. A number is wondrous if and only if it 
eventually reaches 1 through this process.

4 is wondrous because it gets to 1 eventually (after two steps).
15 is wondrous. (Try it out: it takes 17 steps to get there and you 
go as high as 160 in the process, but you eventually get to 16 after 
13 steps and then four more to 1.)

The interesting thing about wondrous numbers is that it's very hard 
to tell in advance how long it will take to get to 1, or if you'll be 
able to get there at all. There is a conjecture that all numbers are 
wondrous, but to my knowledge it hasn't been proven.

If you have a whole lot of time to kill one day, try to find out if 
27 is wondrous. I haven't tried this myself, but as a Tortoise once 
told me, "I don't promise anything.... And I'd advise you to bring 
along a rather long sheet of paper." (Hofstadter p. 401.)

I'd like to add to his warning by saying that some numbers might be 
non-wondrous, so you might need a very long piece of paper indeed.

Hope this helps. Write in with any other questions,

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/