Only 1 and 2

Number Theory Level pending

A positive integer with \(20\) digits is a interesting number if it only has the digits \(1\) and \(2\). (For example, the \(1222212212112121112\) is an interesting number.

Find the maximum value of \(k\), such that there exist \(k\) interesting numbers where every pair of them differ in at least 3 places.

For example we cannot write both \(111\dots1112\) and \(1111\dots1222\), but we can write down \(121212\dots121212\) and \(212121\dots212121\).

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