Thank you.

Actually, not as long as you would think.  I copied the string of numbers into a spreadsheet and then checked the difference between terms and got:

1 4 5 9   0 5  4 6   2 7 0
   3 1 4 -9 5 -1 2 -4 5 -3

Being something of a pi nutcase myself, I noticed the 314 and then that -9 was sort of like a 1 in a bizarro modulo 10 way.  So then I thought, what can I do to turn the -9 into a 1; so I changed the zero to a 10.   So then came the idea that the digits given to us were just the last digits of a series of larger numbers.  Made a new line on the spreadsheet adding however many multiples of 10 were required to keep the series increasing, and got:

1 4 5 9 10 15 24 26 32 37 40
   3 1 4  1   5  9   2  6   5   3

Only later did I notice the hint in the title about cake.
"Problem: You can't have your cake and eat it too."  get it?  Cake?  Pie?  Pi.

As to the philosophical question regarding all series, it would seem the that simplest rule that successfully predicts the series would be the correct one.  Of course the word "simplest" could be open to interpretation, but of the infinite number of rules that might work, we should be able to at least eliminate any rule when there is some other rule that is both simpler and still works.

ps Here is the edited note; the first time, I copied and pasted from a spreadsheet, and the result was lots of raw html; sorry.

Edited on April 15, 2004, 5:49 am

Edited on April 15, 2004, 5:50 am

Posted by Larry on 2004-04-14 23:29:29