Through this problem into GPT4 and here was the response:
Me: Let f: N--> N map a number to the number-of-letters in its name in English
So for example 14 is fourteen so it maps to 7. Lets iterate
14 --> 7 --> 5 --> 4-->4-->4 ...
It is known (perhaps well-known) that this is what ALWAYS happens.
That is, the sequence always becomes all fours eventually. What is the proof?
GPT4:
The proof you're looking for is based on the observation that iterating the
function f(n) eventually leads to a repeating cycle, which in this case is 4.
Let's analyze the possible outcomes of the function to understand why this is the case.
First, we need to define the number of letters in the names of the numbers
1-9, 10-19, and the tens from 20-90:
One, two, three, four, five, six, seven, eight, and nine have
3, 3, 5, 4, 4, 3, 5, 5, and 4 letters, respectively.
Ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen,
eighteen, and nineteen have 3, 6, 6, 8, 8, 7, 7, 9, 8, and 8 letters, respectively.
Twenty, thirty, forty, fifty, sixty, seventy, eighty, and ninety have
6, 6, 5, 5, 5, 7, 6, and 6 letters, respectively.
Now let's consider the range of values that f(n) can take:
The smallest number of letters is 3, for "one" and "two."
The largest possible number of letters for a two-digit number is 10,
for "twenty-eight" or "ninety-eight" (each has 6 + 4 = 10 letters).
Given this, we can analyze the possible outcomes
for each value in the range of 3-10:
f(3) = 5 ("three" has 5 letters)
f(4) = 4 ("four" has 4 letters)
f(5) = 4 ("five" has 4 letters)
f(6) = 3 ("six" has 3 letters)
f(7) = 5 ("seven" has 5 letters)
f(8) = 5 ("eight" has 5 letters)
f(9) = 4 ("nine" has 4 letters)
f(10) = 3 ("ten" has 3 letters)
f values beyond 10 will not be reached since the largest possible value
for a two-digit number is 10, and we already know that f(10) = 3.
From this analysis, we can see that iterating the function f(n) will
eventually lead to the repeating cycle of 4:
3 --> 5 --> 4 --> 4 --> 4 ...
This proves that no matter the starting number,
the sequence will always become all fours eventually.