# binary circle - difficult question

I ran into this question and I'm not really sure how to start.

we are looking at 100 0/1's that are written arround a circle. for a binary sequence $w$, we'll define $n_{w}$ as the number of times $w$ appears arround the circle.

prove that the numbers {$n_{000},n_{001},n_{010},n_{011},n_{100},n_{101},n_{110},n_{111}$} can all be more than 0 (all at once) if and only if:

$$\sum_{w\in \{0,1\}^3}^{}{}n_{w}=100$$ $$n_{100}=n_{001}, n_{011}=n_{110}, n_{101}+n_{001}=n_{010}+n_{011}, n_{110}+n_{010}=n_{100}+n_{101}$$

i would get a clue and not the full answer,

Thank you very much in advance.

• Odd. Unless I misunderstood the problem, 100 zeroes satisfy the second set of conditions but not the first.... – N. S. May 9 '13 at 14:32
• I need it to answer both sets of conditions. – user76508 May 9 '13 at 14:34
• and, it needs to be a proof, i can't use an example – user76508 May 9 '13 at 14:34
• if and only if means that the two sets of conditions are equivalent. So if second is true, first cannot be false.... – N. S. May 9 '13 at 14:35
• gotcha. i'll check it out – user76508 May 9 '13 at 14:36

I think that you need to add the assumption that $w$ is not constant, i.e., that it contains at least one $1$ and at least one $0$. For a HINT, note that $n_{101}+n_{001}$ is the number of subsequences of length $3$ that end with $01$, while $n_{010}+n_{011}$ is the number that begin with $01$. The other equalities are similar, even the first two: $n_{100}=n_{001}$ is equivalent to $n_{000}+n_{100}=n_{000}+n_{001}$.