# Prove that if the starting string has only two digits, then the sequence of strings will eventually repeat. Include an explanation and reasons please.Consider this string of digits: A =...

Prove that if the starting string has only two digits, then the sequence of strings will eventually repeat. Include an explanation and reasons please.

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 0s 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.

Prove that if the starting string has only two digits, then the sequence of strings will eventually repeat.

Please include an explanation and reasons. Thank you!

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### 1 Answer

There are two cases:

(1) Consider the number 11 as the generating string. Then the sequence of derived strings is:

11

0200000000

9010000000

8100000001

7200000010

7110000100

6300000100*

7101001000

6300000100*

And the sequnce begins to repeat. This same sequence will occur for any two digit string where the numbers are the same. Thus 00,11,22,...,99 all will have the derived string consisting of 9 zeros and a 2. The next derived string will then be 9010000000 and from then on will be as given.

**Thus any beginning string of length 2 with repeated digits repeats.**

(2) Assume the numbers in the string are different. Consider 12:

12

0110000000

8200000000

8010000010

7200000010

7110000100

6300000100*

7101001000

6300000100*

And we begin to repeat. This is true of all two digit sequences of different numbers. The first derived string will consist of 8 zeros and 2 1's. The next derived sequence will then be 8200000000 and from then on the sequence is as given.

**Thus any two digit sequence with different numbers will have a sequence of derived strings that repeat.**