This post is inspired by my friends who are driven by the Collatz Conjecture (see prior posts: HOTPO, additional thoughts). The Numberphile Podcast has a short video introducing the Brussels Choice, a problem of sequences. The guest on this episode is the founder of the OEIS (the On-Line Encyclopedia of Integer Sequences), Neil Sloane.
Like the Collatz Conjecture, one starts with any integer and follows simple rules to convert the number to 1 (or other numbers) in a series of steps. Unlike the Collatz Conjecture, not all numbers can be converted to 1. Numbers ending in 5 and 0 cannot be converted to 1 but can be converted to 5.
Rules:
- Any digit or sequence of digits within the number that ends with an even number can be doubled or halved. The other digits are unchanged.
- Any digit or sequence of digits within the number that ends even an odd number can only be doubled. The other digits are unchanged.
Example - start with 6113
Double the 3, the remaining digits are unchanged: 6116
Divide 16 by 2: 618
Divide 8 by 2: 614
Divide 4 by 2: 612
Divide 2 by 2: 611
Divide 6 by 2: 311
Double the final 1: 312
Divide 312 by 2: 156
Double 1: 256 (a power of 2)
Divide 256 by 2 and repeat 7 more times to reach 1.
Example - start with 90 (numbers ending with a 0 or 5 can be converted to 5)
Double 9: 180
Divide 8 by 2: 140
Double 14: 280
Double 28: 560
Double 56: 1120
Divide 12 by 2: 160
Divide 16 by 2: 80
Divide 8 by 2: 40
Divide 4 by 2: 20
Divide 2 by 2: 10
Divide 10 by 2: 5
The site Code Golf, has a challenge to write the code to determine if two numbers are connected by the Brussels Choice.
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