I ran across an interesting post yesterday from Second Effort blog:Second Effort: Heads or Tails #28 -- 7 things
OK -- seven numbers: 1, 7, 10, 13, 19, 23, and 28.I had never heard of Happy Numbers, either. The post reminded me of the Collatz Conjecture. I first heard of this conjecture in Godel, Escher, Bach by Douglas Hofstadter. Hofstadter calls these numbers Wondrous Numbers.
These, it turns out, are the first seven "happy numbers"— something I'd never heard of either before undertaking today's assignment.
Here's how they work:
7^2 = 49
4^2 + 9^2 = 97
9^2 + 7^2 = 130
1^2 + 3^2 + 0^2 = 10
1^2 + 0^2 = 1
Did you get that? Take a number and square it. Break the resulting number into individual digits. Square each digit and add them up. Repeat for the resulting sum... and do so as necessary until (quoting now from the Wikipedia entry) "the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers.
Take any whole number and apply one of these two rules to it: if the number is even, cut it in half, but if odd, multiply it by 3 and add 1. Whatever number results from this, apply one of the two rules to it. Do this again and again and you will notice something odd happening to the numbers that you get.
Cut The Knot has a fun applet to help you explore these numbers.
Want to join? Drop me an e-mail with the subject: Godel-Escher-Bach (click that link and an e-mail message addressed to me with this subject will open in your e-mail client).
Thank you.
Technorati Tags: Collatz Conjecture, Happy Numbers, Godel Escher Bach, Douglas Hofstadter, Paul Erdos, John Allen Paulos, fractal art, Clay Institute, Millenium Problems
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