Conserving contiguity with a transposed progression

[Loren Booda]

Can a sequence of integers from 1 to N be rearranged so that no two neighbors retain their original adjacency?

 

[dodo]

I suppose running it backwards (6,5,4,3,2,1) is not enough for what you meant.

Then, how about 1,3,5,2,4,6.

 

[Loren Booda]

And for any N?

 

[dodo]

Well, for N <= 3 you obviously can't, so the trivial response is "no".

For N >= 5 you only need to list the odds, then the evens.

The interesting case is N=4. If you don't count the endpoints, 2,4,1,3 suffices. If you count them as neighbors, then
- Sequences starting with "1" won't have a place for "2": second and fourth are neighbors to "1", and third place would make it neighbor to wherever you place "3".
- Similar for sequences starting with "4": there's no place for "3" (third place leaves no place for "2").
- Similar when starting with "2", where there is no place for "3" (second and fourth are neighbors to "2", third would leave no place for "4").
- And same for starting with "3", where similarly there is no place for "2" (third place leaves no place for "1").
So with wrap-around, N=4 is a no-no.

 

[blue_raver22]

Yes, it is possible to rearrange a sequence of integers from 1 to N in a way that no two neighbors retain their original adjacency. This is known as a transposed progression.

To achieve this, we can use a mathematical technique called a transposition. This involves swapping two adjacent numbers in the sequence. By repeating this process multiple times, we can rearrange the sequence in a way that breaks the original adjacency.

For example, let's consider the sequence 1, 2, 3, 4, 5, 6, 7. By performing a transposition between 3 and 4, we get the sequence 1, 2, 4, 3, 5, 6, 7. We can continue to perform transpositions between adjacent numbers until we achieve the desired rearrangement.

This technique can be used for any sequence of integers from 1 to N, as long as N is greater than 2. However, it is important to note that this rearrangement may not always be possible. In some cases, the sequence may have a specific pattern or structure that makes it impossible to break the original adjacency. In these cases, it may be necessary to use other mathematical techniques or algorithms to achieve a transposed progression.