# Prove that if the starting string has only two digits, then the sequence will eventually repeat. Include a convincing argument.Consider this string of digits: A=03161011511417191111 It has two 0s,...

Prove that if the starting string has only two digits, then the sequence will eventually repeat. Include a convincing argument.

Consider this string of digits:

A=03161011511417191111

It has two 0s, twelve 1s, zero 2s, and so on.

We construct another string of digits, called B, as follows: write the number of zeros in A, followed by the number of 1s, followed by the number of 2s, and so on until we write the number of 9s. Thus

B=21201111101

String B is called the derived string of A. We now repeat this procedure on B to get its derived string C, then get the derived string of C, and so on to produce a sequence of derived strings.

A=03161011511417191111

B=21201111101

C=2720000000

D=7020000100

E=7110000100

F=6300000100

G=7101001000

H=6300000100

Notice that the last string equals a previous string so the sequence of derived strings will now repeat.

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### 3 Answers

if the starting string has 2 digits, it is either 2 times the same digit or 2 different digits.

In this case there will be either 1 2 in the sequence or 2 1

**If there is 1 2,** then the third string will be 0010000

Starting to the fourth one, the sequences will be 0100000000 (1 1 and 0 other digits) all the time therefore it repeats....

If there are 2 1, then the third sequence will be 0200000000

then the fourth :0010000000

Starting to the fifth, the sequences will always be 0100000000 therefore it repeats.

**In conclusion, the sequences repeats after the 5th sequence for any 2 digits initial sequence.**

Unfortunately your answer didn't make much sense to me, and I couldn't answer my question properly. What I need is an elaboration on that answer and more explanation, I really need to understand the answer properly! Thank you!

While you're at it could you maybe post it instead on my question here:

http://www.enotes.com/math/q-and-a/prove-that-starting-string-has-only-two-digits-331793

So that I can get an update when you post an answer (if you do). Thanks so much!

Cosinusix, sorry, I didn't quite understand your answer to this question, could you explain / elaborate on it more? Thank you so much!