$x_1 x_2 x_3 x_4 + x_2 x_3 x_4 x_5 +......+ x_n x_1 x_2 x_3 = 0$ then what is $n$? Can anyone please help me to understand what is the following problem saying?[!

Each of the numbers $x_1,x_2,\cdots,x_n,n>4$, is equal to $1$ or $-1$. Suppose 
  $$x_1x_2x_3x_4+x_2x_3x_4x_5+\cdots +x_nx_1x_2x_3=0$$
  then, 
  $(1)$ $n$ is even, $(2)$ $n$ is odd, $(3)$ $n$ is odd multiple of $3$, $(4)$ $n$ is prime.

I am really having a hard time to understand this problem. What I understood is I have to find a natural number $n$ for which the equation holds whatever the values of $x_i$ s are. That is impossible. That's why it seems that some conditions on $x_i$ s are necessary.

The first part of the problem is:

"Each of the numbers $x_1,\ \cdots, x_n$ is equal to 1 or -1".

Thus, the restriction on the values of $x_i$ is that each and every one must be one of either $1$ or $-1$, and nothing else. They don't all have to be the same number, though they could be, but the only numbers allowed for each one individually are either $1$ or $-1$. So, if we had, say, $n = 5$, then
$$(1)(-1)(1)(1)(1)$$
would be an okay product, but
$$(1)(2)(1)(1)(1)$$
would not be, since now one of the $x_i$s (which one depends on where in the "cycle" we are imagining this product as representing) is $2$, and $2$ is not a value we are allowed to use.
Thus, you can then ask:


*

*Given that, what are the only possible values for any products formed from such $x_i$?

*After you find that, what must be true about the values of the specific products given, so that their sum is 0?


This should imply something about $n$, as for only certain $n$ values will you be able to make (2) work out with the constraint of allowed values by (1).
