<p>You are given a positive integer <code>n</code>.</p>

<p>A binary string <code>x</code> is <strong>valid</strong> if all <span data-keyword="substring-nonempty">substrings</span> of <code>x</code> of length 2 contain <strong>at least</strong> one <code>&quot;1&quot;</code>.</p>

<p>Return all <strong>valid</strong> strings with length <code>n</code><strong>, </strong>in <em>any</em> order.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 3</span></p>

<p><strong>Output:</strong> <span class="example-io">[&quot;010&quot;,&quot;011&quot;,&quot;101&quot;,&quot;110&quot;,&quot;111&quot;]</span></p>

<p><strong>Explanation:</strong></p>

<p>The valid strings of length 3 are: <code>&quot;010&quot;</code>, <code>&quot;011&quot;</code>, <code>&quot;101&quot;</code>, <code>&quot;110&quot;</code>, and <code>&quot;111&quot;</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 1</span></p>

<p><strong>Output:</strong> <span class="example-io">[&quot;0&quot;,&quot;1&quot;]</span></p>

<p><strong>Explanation:</strong></p>

<p>The valid strings of length 1 are: <code>&quot;0&quot;</code> and <code>&quot;1&quot;</code>.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 18</code></li>
</ul>

class Solution:
    def validStrings(self, n: int) -> List[str]:
        if n == 1:
            return ["0", '1']
        reached = set()

        s = "1"*n
        def helper(s):
            #print(s)
            if s in reached:
                return
            reached.add(s)
            lis = list(s) 
            for i in range(n):
                if not lis[i] == "1":
                    continue
               #print(i, lis)
                if ((i == 0 and lis[i+1] == "1") 
                    or (i == n-1 and lis[i-1] == "1") 
                    or ( (0<=i<n-1) and (lis[i-1] == "1" and lis[i+1] == "1"))):
                        lis[i] = "0"
                        #print(lis, i)
                        helper("".join(lis))
                        lis[i] = "1"
        helper(s)
        return list(reached)

        
83