How many terms in this sequence are squares? $$1, 11, 111, 1111, \cdots $$
The answer is $\boxed{1}$.
All these terms are odd. In order for any one to be a square, it must be congruent to $1$ modulo $4$. Only the first term $1$ satisfies this condition. And clearly, $1$ is a square. Hence, the answer is $1$.