More Pi-Position trivia:
* The smallest integer that maps to a 4-digit number is
However, I think we can give up on multiple 0 patterns. They just didn't prove to be good ones.
Maybe we should have expected that 0 would be a troublemaker. After all, of the digits 0 through 9, it's the one that first occurs the latest in pi (starting in position 32), well after the other digits have appeared. In fact, the first 0 appears after every other digit has appeared at least twice. 0 is either lazy or shy!
* The smallest integer that maps to a 4-digit number is
103 -> 3486
which means that the sequence 1 0 3 does not occur for more than 3000 digits into pi. But it is quickly reined back in:103 -> 3486 -> 265 -> 6 -> 7 -> 13 -> 110 -> 174 -> 155 -> 314 -> 0
* I found another loop, starting from 40. It goes from 2-digit numbers to 3-digit numbers to 4-digit numbers to 5-digit numbers, but manages to return back to 2 with a few hops between 4 and 5. That's unusual already, but it also hits the number it started with!40 -> 70 -> 96 -> 180 -> 3664 -> 24717 -> 15492 -> 84198 -> 65489 -> 3725 -> 16974 -> 41702 -> 3788 -> 5757 -> 1958 -> 14609 -> 62892 -> 44745 -> 9385 -> 169 -> 40 -> ...
* Even though we start from an integer, we are looking for sequences of digits. That means that we'll get different results if we add leading zeros. We already know that0 -> 32 -> 15 -> 3 -> 0 -> ...
so let's try double zero (is that a James Bond reference?), triple zero, and quadruple zero:00 -> 307 -> 64 -> 22 -> 135 -> 2727 -> 13232 -> 202354 -> 665958 -> 1653738 -> 2634420 -> 19322923 -> 81263807 -> 1522737 -> 1799951 -> 23495684 -> 20737631 -> 181074916 -> out of range
000 -> 601 -> 1217 -> 11077 -> 170359 -> 728783 -> 1066817 -> 5265352 -> 6300025 -> 17973511 -> 15808149 -> 157446283 -> out of range
(Notice that the sequence for triple zero passes through 6300025, which contains a triple zero.)
0000 -> 13390 -> 1985 -> 11065 -> 333640 -> 463664 -> 1053702 -> 7941454 -> 11567006 -> 69921034 -> 86863832 -> 120195172 -> out of range
"Out of range" means the pi-search site I'm using didn't find it in the first 200 million digits of pi. That tells us nothing about what happens later; the sequence could keep growing to larger numbers, return to smaller numbers, or eventually loop.000 -> 601 -> 1217 -> 11077 -> 170359 -> 728783 -> 1066817 -> 5265352 -> 6300025 -> 17973511 -> 15808149 -> 157446283 -> out of range
(Notice that the sequence for triple zero passes through 6300025, which contains a triple zero.)
0000 -> 13390 -> 1985 -> 11065 -> 333640 -> 463664 -> 1053702 -> 7941454 -> 11567006 -> 69921034 -> 86863832 -> 120195172 -> out of range
However, I think we can give up on multiple 0 patterns. They just didn't prove to be good ones.
Maybe we should have expected that 0 would be a troublemaker. After all, of the digits 0 through 9, it's the one that first occurs the latest in pi (starting in position 32), well after the other digits have appeared. In fact, the first 0 appears after every other digit has appeared at least twice. 0 is either lazy or shy!
