A sequence is called green if formed of digits 0,1,2 (digits can be repeated) A sequence is called original if there is no 3 digits that are repeated twice :
An example for non original sequence: 12121, such that 12121 and 12121 are repeatedly
Prove that for every sequence of 30 and up are non original:
My attempt is, we assume in contrary that for every sequence of max 29 is non original. if the assumption is wrong then the statement for every sequence of 30 and up is non original
we have $3^3 = 27$ possibilities to form 3 digits of 0,2,1 and we have set of 29 holes to fill, i don't know how to continue from here
In a sequence of $30$ digits, you have $28$ subsequences of three consecutives digits (the one beginning at the first digit, the one beginning at the second digit, ..., the one begining at the $28$th digit).
But there are only $27$ possible green sequences of three digits. That means that two of the $28$ sequences are the same.