SESNs: Self-Erasure Surviving Numbers & memories of childhood math games
I enjoy algorithmic/computational number puzzles. When I was a kid (age 9 or 10?) I checked out a book from the public library that, among other things, had a simple game you could play on paper or with a calculator:
- Pick a whole number > 0
- If it is odd, multiply it by 3 and add 1
- If it is even, divide it by 2
The book said (IIRC) that all numbers would eventually reach 1. For example, the unlucky number 13 reaches 1 in 9 steps:
- Start with 13.
- (13 x 3) + 1 = 40
- 40/2 = 20
- 20/2 = 10
- 10/2 = 5
- (5 x 3) + 1 = 16
- 16/2 = 8
- 8/2 = 4
- 4/2 = 2
- 2/2 = 1
As a child I didn't know that this game had a real name other than "3x+1". It's actually the Collatz Conjecture, and it is an unsolved problem in number theory as to whether every number will eventually reach 1 or not.
A few years ago I produced some computer generated art based on the 3x+1 problem.
But all this has just been by way of introduction to Eric Angelini's rec.puzzles posting this morning about Self-erasing numbers. In his own words:
Take an integer like 36, for example. Concatenate an infinite amount of copies. You get: 363636363636363636363636363636... - read the leftmost digit -"3"-, - jump *over* 3 digits (to the right), land on (3) and erase it: 3636(3)6363636363636363636363636... ^ - read the leftmost unread digit, jump only visible digits, erase: 3636(3)6363(6)36363636363636363636... ^^ - repeat until you see a substring like this [...(a)36(b)...] [(a) and (b) are erased digits - "36" is the integer you are, testing]: bingo, you have found a "Self-erasure survi- ving number" (SESN): "36" is such a number: 3636(3)63(6)3(6)36(3)(6)3(6)36363636363636... ^^^^ ^^ .. <-- hit This erasing technique gives sometimes quite complicated pat- terns. "16", for instance, is not a SESN -- but it takes a while to see: 16(1)616(1)61(6)1(6)(1)6(1)(6)1(6)1(6)1(6)1(6)(1)6(1)616(1)61(6)1(6) ^^ ^^^ ^^ ^ ^ ^ ^ ^ ^ ^ ^^^ ^^ ^ |_______________________________________________| recurrent pattern The first SESN I have found by hand are: 0 1 2 3 4 5 6 7 8 9 10 20 23 24 25 26 27 28 29 30 32 36 37 38 39 40 42 ... [BTW, reading "0" means erasing the closest visible digit immediately to the right] No SESN > 10 begins with "1" -- see why? No SESN > 299 begins with "2", etc. The sequence is finite, thus. Last term?
I wrote a program to compute whether a number is a Self-Erasure Surviving Number tonight—a fun diversion.
Though, either I have a bug, or I don't see why a number beginning with 1, which is greater than 10, can't be an SESN... it would appear that 114 is a valid SESN. Consider the output from my program (which adopts the notation used by Eric) for ./SESN.tcl 114:
114114114114114114114114114114114114114114114114114114114114 11(4)114114114114114114114114114114114114114114114114114114114 ^ 11(4)1(1)4114114114114114114114114114114114114114114114114114114 ^^ 11(4)1(1)4(1)14114114114114114114114114114114114114114114114114114 ^^ ^ 11(4)1(1)4(1)1411(4)114114114114114114114114114114114114114114114114 ^^ ^ ^ 11(4)1(1)4(1)14(1)1(4)114114114114114114114114114114114114114114114114 ^^ ^ ^ ^ 11(4)1(1)4(1)14(1)1(4)114(1)14114114114114114114114114114114114114114114 ^^ ^ ^ ^^ !!! 114 is a Self-Erasure Surviving Number (SESN)!
Time for bed; there will be time to ponder 114 more deeply during the commute.
Oh, and if you want to play the 3x+1 game sometime—perhaps during a boring meeting, or when you are stuck in traffic and have a lot of time to kill—start with 27...
—Michael A. Cleverly
Thursday, July 07, 2005 at 23:53
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Wed, 15 Jul 2020, 06:11