Thursday, 15 March 2012

Pi Tag

3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862
8034825342117067982148086513282306647093844609550582231725359408128481117450284102701
9385211055596446229489549303819644288109756659334461284756482337867831652712019091456
4856692346034861045432664821339360726024914127372458700660631558817488152092096282925
4091715364367892590360011330530548820466521384146951941511609...

If you pick some number, any number, and look for it somewhere in the sequence of digits of pi, it will show up... I can't prove that, but I believe it.... For example, I was born in October, the tenth month.  The digits 10 show up starting at the forty-ninth decimal place of pi  (I ignore the 3 before the decimal point just to avoid counting complexities).

Now if you take the location as a new number, and look for the place where that occurs in pi, you create a sequence of numbers that must do one of three things.  Either they keep going, jumping up and down maybe and go off to infinity (or at least to a really big number), or they cycle through the same set of numbers, or they simply go to a single  number which happens to be in the position of its numeral value.... like 1.  The first number after the decimal point is one, so we just keep going  back to one...

But if you start with four, which is in position two, and two is in position six, the string looks like this
4  ,  2, 6, 7, 13, 110, 174, 155, 314, 2120 , 5360... eventually I got to 119,546 and I give up on that one... maybe it actually gets smaller... One of those big numbers might turn up in place 141 or 1415 or 14159 and each of those would take us back to position one, but I'm willing to say, for the moment, if they go past 1000, they've gone out of bounds... Folks with big computers and the patience to program them may find a way to take this out to a plethora of digits, but I'm working with one crunch at a time...

As you go through the simple numbers it seems that all of them run off to big numbers, then 14 shows up and it goes to position one.

But if  you try 19, you find it is in position 37.  37 turns up in position 46, but 46 occurs in position number 19, and we have a cycle... trivia alert, 19 is the smallest number in Pi Tag that has a non-trivial (goes to one and repeats ones forever) cycle.

I'm really hoping I can find a two -cycle... a number that gives a number that brings us back where we started... a  leads to b, leads back to a.... Sort of a buddy system... there could be lots of them...

If you want to play around with this, there is a pretty nice web site that lets you put in any number and see what position it shows up in... and it handles really big stuff, I put in my social security number just to see if it's location was bigger than the number or smaller...turns out it only goes up to 200 x 106 and my number didn't show up by then...
The page is  here  if you want to give it a go.

It's pretty easy to check because as soon as a number goes to a number lower than itself, you know what it will do (you've already checked that number)  for example, 23 is in position 16, but I know 16 goes off to above 1000, so stop and go to the next number...

And if you find a two-cycle before I do, or any other cycles, drop me a note..... and now back to work...