There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$?
Problem
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Tags: algebra
Socoobo
16.11.2021 06:33
I claim that the maximal value of $n$ is $15$. One possible sequence of $n=15$ that satisfies the conditions is
\[-1,-1,-1,-1,-1,1,1,1,1,1,-1,-1,-1,-1,-1.\]
Now we prove $n$ cannot be $\geq16$. If we let the terms be $a_i$ for $i\geq1$, then we have $a_i=a_{j}$ if and only if $i\equiv j\pmod{10}$, due to the condition that the sum of every $10$ neighboring numbers is $0$. Without loss of generality, let $a_1=a_{11}=-1$. Consider $a_i$ for $i\geq12$. If any $a_i$ is $1$, then
\[\sum_{k=i-11}^ia_k=0\]which contradicts the condition that the sum of every $12$ numbers cannot be $0$. Thus, we have that all $a_i=-1$ for $i\geq12$. But if $n\geq16$, then we will have a consecutive string of $6$ or more $-1$ at the end, so that violates the condition that the sum of all $10$ consecutive terms is $0$. Thus $n\leq15$ so $15$ is the maximum.