Friday, June 22, 2012

153

In John 21, John recalls a time when they were fishing, and Jesus asked them to throw their nets on the other side of the boat, and when they obeyed they were unable to haul the nets in. This was the second time this had happened, one of the times Jesus appeared to the disciples after his resurrection.

I always loved the little detail John throws in verse 11: there were 153 fish.

That particular number always seemed a little odd to me. (yes, pun intended... you can stop laughing and start reading again anytime...)

Besides being odd, 153 has a lot of other interesting properties, and the one I want to describe is a remarkable fact that 1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153.  How remarkable is that?! When you cube the digits of a number, and add them up, you get the same number back! Now, does that always happen? That is, when you do this little operation--I'm going to call it a "cubic sum"-- do you always get your number back?

That's easy enough to check -- lets find the "cubic sum" of 972.
9^3 = 729
7^3 = 343
+ 2^3 = 8
1080

Nope. In fact, 153 is one of very few numbers that equals the "cubic sum" of its digits. Wikipedia calls that a "Narcissistic Number" but I call it cool. The only other numbers with that property (for cubes at least) are 0, 1, 307, 371, and 407.  Lets check 371 a second:
3^3 = 27
7^3 = 343
+ 1^3 = 1
371

Now if that was all that I knew on the topic, that would be cool enough, and your day would be better for having read it, but there's one more fact about 153 and these "cubic sums" that needs to be told.  Lets go back to our first effort of 972.  When we found its cubic sum, it gave us 1080, and we were bummed that we didn't get what we were hoping for. Lets find 1080's cubic sum:
1^3 = 1
0^3 = 0
8^3 = 512
+  0^3 = 0
513
So what?  Well, let's keep going... find the cubic sum of 513 and you get ... drum roll please ... 153!  I have a feeling that's why these numbers are called narcissistic -- because this process often leads back to themselves, even when starting with other numbers!

Now, not every number chosen at random leads to 153, here's a chain I found that leads to 370, having started with the number 259.

259 --> 862 --> 736 --> 586 --> 853 --> 664 --> 496 --> 1009 --> 730 --> 370 --> 370 ...

524 --> 197 --> 1073 --> 371.

In fact, most of the numbers I initially tried went to 371 -- perhaps its even more narcissistic than 153?    Actually, I was just unlucky in the numbers I tried -- because every multiple of 3 will cycle around to 153. I had forgotten that fact... Try 51 = 17*3  and you'll find:
51 --> 126 --> 225 --> 141 --> 66 --> 432
--> 99 --> 1458 --> 702 --> 351 --> 153

I know its even possible to create a loop of numbers that just cycle among themselves, like a clique one such clique of numbers is 133 --> 55 --> 250 --> 133 ...