Pigeonhole Principle: Find Repeating Digits in T={0,1,2}^11

[AkilMAI]

Let T={0,1,2} so that T^11 represents the set of all strings of eleven digits of T.
a)for every T=T_1T_2...T_11 show that there is a pair T_iT_i+1 of consecutive digits that is repeated using the pigenhole principle.
b)Find a string from T of 10 digits where no repetition exists.
c)Find with justification a positive integer N such that every string of N digits of T contains a repeated triple T_iT_i+1T_i+2.

Answers:
a)Since there are 9 possible pairs and 11 digits one pair will be contained at least once in the string.Is this correct?
b)0011220102.
c)I don't know.How should I proceed?

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[vela]

Let T={0,1,2} so that T^11 represents the set of all strings of eleven digits of T.
a)for every T=T_1T_2...T_11 show that there is a pair T_iT_i+1 of consecutive digits that is repeated using the pigenhole principle.
b)Find a string from T of 10 digits where no repetition exists.
c)Find with justification a positive integer N such that every string of N digits of T contains a repeated triple T_iT_i+1T_i+2.

Answers:
a)Since there are 9 possible pairs and 11 digits one pair will be contained at least once in the string.Is this correct?

Yeah, that's right.

b)0011220102.

Try again!

c)I don't know.How should I proceed?

Think about why you can write a 10-digit number with no pairs repeating but not an 11-digit number. How would you generalize this to the case of triples?

 

[AkilMAI]

b)2001022112
c)There are 27 different triples
a string of n digist must have n-2 triples in it,we are substracting the last 2 digits.
So 28+2=30 will work.(or I could just look in terms of 27+another triple(3)=30)

 

[vela]

Good job!