Langford series

Notice anything special about the following sequence?

8 6 10 3 1 11 1 3 6 8 12 9 7 10 4 2 5 11 2 4 7 9 5 12

Each of the numbers 1 through 12 appear twice. Between the two 1s there is one number. Between the two 2s there are two numbers. Between the two 3s there are three numbers, etc.

Langford’s problem of order n is to arrange two copies of the integers 1 through n so that there are k numbers between the two ks. This problem has a solution if and only if n is congruent to 0 or 3 mod 4.

You can find much more on Langford’s problem here.

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