NIaS: Thue-Morse

Probability Level 3

Consider the following function applied on a sequence of numbers:

First, group together identical numbers in a row in the sequence and count them. For example, for the sequence $1,2,3,3,2,2,2,1,1,2,$ group it as follows and count the number of elements in each group: $\underbrace { (1) }_{ 1 } ,\underbrace { (2) }_{ 1 },\underbrace { (3,3) }_{ 2 } ,\underbrace { (2,2,2) }_{ 3 } ,\underbrace { (1,1) }_{ 2 } ,\underbrace { (2) }_{ 1 }.$
Then, take the new sequence created by the numbers counted (the ones under the braces): $1,1,2,3,2,1.$

Now, apply this function to the infinite Thue-Morse sequence multiple times.

What is the largest number you can get in one of the altered sequences?

1 2 3 5 14 26 58 It's infinitely large

1 solution

Louis Ullman
Apr 20, 2018

Applying the function: $1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,2,2,1,1,2,2,1,2,1,1,2...\\ 1,2,1,1,2,2,2,1,1,2,1,1,2,1,1,2,2,2,1,1,2...\\ 1,1,2,3,2,1,2,1,2,3,2,1...\\ 2,1,1,1,1,1,1,1,1,1,1...\\ 2,\boxed { \infty } ,2$

NIaS: Thue-Morse

1 solution

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