We answer last weeks puzzle, discuss the law of small numbers and ask again what is the smallest positive counting number that Google can’t find?
As a bonus we discuss the Fibonacci Substitution Sequence and close out with some Fibonacci Substitution Sequence Drumming by Akio Hizume.
To explore the sequence, we apply the following rules: replace each 0 with 001; replace each 1 with 01.
Beginning with 0, we next have 001. Next, applying our rules again, we have 001 001 01. In this way we obtain:
Amusingly the numbers of 0’s and 1’s at each step are a consecutive pair of fibonacci numbers, and the ratio of 0’s to 1’s, in the limit, is the famous golden ratio.
EDIT: We initially miswrote the substitution as
replace each 0 with 01; replace each 1 with 001, giving
Stevestyle caught the error in the comments below