MHB Is Two-Colour Bead Necklace Arrangement Possible?

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A necklace with 64 beads in 8 different colors can be arranged to show every possible succession of two colors exactly once, which is known as a De Bruijn sequence. This specific sequence is referred to as B(8,2) and can be visualized with colors labeled from 0 to 7. The arrangement forms a cyclic sequence, allowing traversal in either direction. The existence of such sequences can be proven through established mathematical constructions and references, such as West's Introduction to Graph Theory. Therefore, it is indeed possible to create the desired necklace arrangement.
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Hello! (Wave)

We have a necklace with $64$ beads from $8$ different colours. Is it possible that they are put in such a way that if we go along the necklace in one direction (in any of the two) then we see all the possible successions of two colours exactly once?
 
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evinda said:
Hello! (Wave)

We have a necklace with $64$ beads from $8$ different colours. Is it possible that they are put in such a way that if we go along the necklace in one direction (in any of the two) then we see all the possible successions of two colours exactly once?
This is called a De Bruijn sequence. (I'm assuming that the necklace is joined up at the ends so as to form a cyclic sequence of beads.)

The sequence that you want is the De Bruijn sequence $B(8,2)$. If you want to see an example, here it is (with the colours labelled $0$ to $7$).

[sp]http://www.hakank.org/comb/debruijn.cgi?k=8&n=2&submit=Ok[/sp]
 
Opalg said:
This is called a De Bruijn sequence. (I'm assuming that the necklace is joined up at the ends so as to form a cyclic sequence of beads.)

The sequence that you want is the De Bruijn sequence $B(8,2)$. If you want to see an example, here it is (with the colours labelled $0$ to $7$).

[sp]http://www.hakank.org/comb/debruijn.cgi?k=8&n=2&submit=Ok[/sp]

And how could we prove that such a sequence exists? (Thinking)
 
evinda said:
And how could we prove that such a sequence exists? (Thinking)
The construction of $B(2, 4)$ is illustrated in the wiki article. That should explain how to construct such sequences. You can see the proof of the general statement in West's Introduction to Graph Theory. Look for "De-Bruijn Cycles" in the index.
 

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