# How can $ww = www$ hold for any word $w$?

Speaking in terms of automata and regular languages, how would it be possible for a string repeating some $w$ twice equal a string repeating that same $w$ thrice? That is, why is the language

$\qquad L = \{w \in \Sigma^ * \mid ww = www\}$

not empty? The only thing I can think of is $w = abab, ww = abababab, www = abababababab$, but I don't think this is correct.

• Is this homework? Are you sure you read the question right? The language $L$ is not empty, but $L = \{\epsilon\}$. – Pål GD Feb 2 '13 at 13:51

The only way that $w w = w w w$ is that $w = \epsilon$. Algebra of strings (for mathematician types, the free monoid on $\Sigma$) isn't that different from multiplication...
A string of length $2$ cannot be equal to a string of length $3$. But a string constructed by concatenation of $2$ strings may be equal to a string constructed by concatenating $3$ strings. The empty string (denoted by $\epsilon$) is of length = $0$ and it's a member in $\Sigma^*$. If $w$ is chosen to be the empty string, then $\epsilon \epsilon = \epsilon \epsilon \epsilon$.
This is true because for any string $w$, $w\epsilon$ = $\epsilon w$ = $w$.
It can be possible because of the empty string $\epsilon$ which is a part of $\Sigma^*$