For clarification, I am very bad at maths and the logic usually goes right over my head, however I am studying a reasonably high level of maths because I am a software and game development student.
I was posed this problem: "How can you solve the rotating drum problem (de Bruijn sequence) for 64 segments?"
I have spent hours trying to comprehend how the rotating drum problem works, and I am slowly coming to terms with it theoretically. However, surely when you add more segments it becomes a different problem because of repeating binary sequences? Or have I completely misunderstood the problem?
If anyone can help me to understand the rotating drum problem in any respects, or give me some guidance as to how to go about solving for 64 segments, it would be very much appreciated!
To solve the rotating drum problem in this case is to find a binary de Bruijn sequence for length-$6$, i.e. a cycle of $64$ bits in which every possible start point gives every possible $6$-bit word.
There's a cheap way to get a de Bruijn sequence for words of length $n$: concatenate in lexicographic order all Lyndon words whose length divides $n$. In this case $n=6$, so the relevant Lyndon words are $$0,1,01,001,011,000001,000011,000101,000111,001011,001101,001111,010111,011111$$ The de Bruijn sequence constructed from them is $$0\ 000001\ 000011\ 000101\ 000111\ 001\ 001011\ 001101\ 001111\ 01\ 010111\ 011\ 011111\ 1$$ or, as one long string (wrapped into a circle of course),