How to get a specific sequence like this?

Question

There are 100 numbers:

1, 1, 2, 2, 3, 3,.. 50, 50.

How can I get a sequence which has one number between the two 1s, two numbers between the two 2s, three numbers between the two 3s,.. and fifty numbers between the two 50s using the hundred numbers?

Does anyone have a better idea than brute force? Or prove it there's no solution when n = 50.

Answer 1

The Problem: Langford Pairing

The problem is known as Langford Pairing. From Wikipedia:

These sequences are named after C. Dudley Langford, who posed the problem of constructing them in 1958. As Knuth¹ describes, the problem of listing ALL Langford pairings for a given N can be solved as an instance of the exact cover problem (one of Karp's 21 NP-complete problem), but for large N the number of solutions can be calculated more efficiently by algebraic methods.

¹: _{The Art of Computer Programming, IV, Fascicle 0: Introduction to Combinatorial Algorithms and Boolean Functions}

It's worth noting that there is no solution for N = 50. Solutions only exist for N = 4k or N = 4k - 1. This is proven in a paper by Roy A. Davies (On Langford's Problem II, Math. Gaz. 43, 253-255, 1959), which also gives the pattern to construct a single solution for any feasible N.

Related links

• John Miller's page on Langford's Problem
  • A Perl program based on Roy A. Davies' pattern
  • Dave Moore's pattern
• mathworld.wolfram - Langford's Problem
• A CSP model for Microsoft Solver Foundation

---

Brute force in Java

Here are some solutions that my quick and dirty brute force program was able to find for N < 100. Nearly all of these were found in under a second.

 3 [3, 1, 2, 1, 3, 2]
 4 [4, 1, 3, 1, 2, 4, 3, 2]
 7 [7, 3, 6, 2, 5, 3, 2, 4, 7, 6, 5, 1, 4, 1]
 8 [8, 3, 7, 2, 6, 3, 2, 4, 5, 8, 7, 6, 4, 1, 5, 1]
11 [11, 6, 10, 2, 9, 3, 2, 8, 6, 3, 7, 5, 11, 10, 9, 4, 8, 5, 7, 1, 4, 1]
12 [12, 10, 11, 6, 4, 5, 9, 7, 8, 4, 6, 5, 10, 12, 11, 7, 9, 8, 3, 1, 2, 1, 3, 2]
15 [15, 13, 14, 8, 5, 12, 7, 11, 4, 10, 5, 9, 8, 4, 7, 13, 15, 14, 12, 11, 10, 9, 6, 3, 1, 2, 1, 3, 2, 6]
16 [16, 14, 15, 9, 7, 13, 3, 12, 6, 11, 3, 10, 7, 9, 8, 6, 14, 16, 15, 13, 12, 11, 10, 8, 5, 2, 4, 1, 2, 1, 5, 4]
19 [19, 17, 18, 14, 8, 16, 9, 15, 6, 1, 13, 1, 12, 8, 11, 6, 9, 10, 14, 17, 19, 18, 16, 15, 13, 12, 11, 7, 10, 3, 5, 2, 4, 3, 2, 7, 5, 4]
20 [20, 18, 19, 15, 11, 17, 10, 16, 9, 5, 14, 1, 13, 1, 12, 5, 11, 10, 9, 15, 18, 20, 19, 17, 16, 14, 13, 12, 8, 4, 7, 3, 6, 2, 4, 3, 2, 8, 7, 6]
23 [23, 21, 22, 18, 16, 20, 12, 19, 11, 8, 17, 4, 1, 15, 1, 14, 4, 13, 8, 12, 11, 16, 18, 21, 23, 22, 20, 19, 17, 15, 14, 13, 10, 7, 9, 3, 5, 2, 6, 3, 2, 7, 5, 10, 9, 6]
24 [24, 22, 23, 19, 17, 21, 13, 20, 10, 8, 18, 4, 1, 16, 1, 15, 4, 14, 8, 10, 13, 12, 17, 19, 22, 24, 23, 21, 20, 18, 16, 15, 14, 11, 12, 7, 9, 3, 5, 2, 6, 3, 2, 7, 5, 11, 9, 6]
27 [27, 25, 26, 22, 20, 24, 17, 23, 12, 13, 21, 7, 4, 19, 1, 18, 1, 4, 16, 7, 15, 12, 14, 13, 17, 20, 22, 25, 27, 26, 24, 23, 21, 19, 18, 16, 15, 14, 11, 9, 10, 5, 2, 8, 3, 2, 6, 5, 3, 9, 11, 10, 8, 6]
28 [28, 26, 27, 23, 21, 25, 18, 24, 15, 13, 22, 10, 6, 20, 1, 19, 1, 3, 17, 6, 16, 3, 10, 13, 15, 18, 21, 23, 26, 28, 27, 25, 24, 22, 20, 19, 17, 16, 14, 12, 9, 7, 11, 4, 2, 5, 8, 2, 4, 7, 9, 5, 12, 14, 11, 8]
31 [31, 29, 30, 26, 24, 28, 21, 27, 18, 16, 25, 13, 11, 23, 6, 22, 5, 1, 20, 1, 19, 6, 5, 17, 11, 13, 16, 18, 21, 24, 26, 29, 31, 30, 28, 27, 25, 23, 22, 20, 19, 17, 15, 12, 14, 9, 10, 2, 3, 4, 2, 8, 3, 7, 4, 9, 12, 10, 15, 14, 8, 7]
32 [32, 30, 31, 27, 25, 29, 22, 28, 19, 17, 26, 13, 11, 24, 6, 23, 5, 1, 21, 1, 20, 6, 5, 18, 11, 13, 16, 17, 19, 22, 25, 27, 30, 32, 31, 29, 28, 26, 24, 23, 21, 20, 18, 16, 15, 12, 14, 9, 10, 2, 3, 4, 2, 8, 3, 7, 4, 9, 12, 10, 15, 14, 8, 7]
35 [35, 33, 34, 30, 28, 32, 25, 31, 22, 20, 29, 17, 14, 27, 10, 26, 5, 6, 24, 1, 23, 1, 5, 21, 6, 10, 19, 14, 18, 17, 20, 22, 25, 28, 30, 33, 35, 34, 32, 31, 29, 27, 26, 24, 23, 21, 19, 18, 16, 13, 15, 12, 9, 4, 2, 11, 3, 2, 4, 8, 3, 7, 9, 13, 12, 16, 15, 11, 8, 7]
36 [36, 34, 35, 31, 29, 33, 26, 32, 23, 21, 30, 17, 14, 28, 10, 27, 5, 6, 25, 1, 24, 1, 5, 22, 6, 10, 20, 14, 19, 17, 18, 21, 23, 26, 29, 31, 34, 36, 35, 33, 32, 30, 28, 27, 25, 24, 22, 20, 19, 18, 16, 13, 15, 12, 9, 4, 2, 11, 3, 2, 4, 8, 3, 7, 9, 13, 12, 16, 15, 11, 8, 7]
40 [40, 38, 39, 35, 33, 37, 30, 36, 27, 25, 34, 22, 20, 32, 17, 31, 13, 11, 29, 7, 28, 3, 1, 26, 1, 3, 24, 7, 23, 11, 13, 21, 17, 20, 22, 25, 27, 30, 33, 35, 38, 40, 39, 37, 36, 34, 32, 31, 29, 28, 26, 24, 23, 21, 19, 16, 18, 15, 12, 4, 6, 14, 2, 5, 4, 2, 10, 6, 9, 5, 8, 12, 16, 15, 19, 18, 14, 10, 9, 8]
43 [43, 41, 42, 38, 36, 40, 33, 39, 30, 28, 37, 25, 23, 35, 20, 34, 16, 14, 32, 5, 31, 8, 6, 29, 2, 5, 27, 2, 26, 6, 8, 24, 14, 16, 22, 20, 23, 25, 28, 30, 33, 36, 38, 41, 43, 42, 40, 39, 37, 35, 34, 32, 31, 29, 27, 26, 24, 22, 21, 19, 17, 15, 18, 12, 7, 1, 3, 1, 13, 4, 3, 11, 7, 10, 4, 9, 12, 15, 17, 19, 21, 18, 13, 11, 10, 9]
44 [44, 42, 43, 39, 37, 41, 34, 40, 31, 29, 38, 26, 24, 36, 20, 35, 16, 14, 33, 5, 32, 8, 6, 30, 2, 5, 28, 2, 27, 6, 8, 25, 14, 16, 23, 20, 22, 24, 26, 29, 31, 34, 37, 39, 42, 44, 43, 41, 40, 38, 36, 35, 33, 32, 30, 28, 27, 25, 23, 22, 21, 19, 17, 15, 18, 12, 7, 1, 3, 1, 13, 4, 3, 11, 7, 10, 4, 9, 12, 15, 17, 19, 21, 18, 13, 11, 10, 9]
52 [52, 50, 51, 47, 45, 49, 42, 48, 39, 37, 46, 34, 32, 44, 29, 43, 26, 24, 41, 20, 40, 16, 14, 38, 7, 9, 36, 2, 35, 3, 2, 33, 7, 3, 31, 9, 30, 14, 16, 28, 20, 27, 24, 26, 29, 32, 34, 37, 39, 42, 45, 47, 50, 52, 51, 49, 48, 46, 44, 43, 41, 40, 38, 36, 35, 33, 31, 30, 28, 27, 25, 23, 21, 19, 22, 15, 8, 6, 1, 18, 1, 17, 4, 5, 6, 8, 13, 4, 12, 5, 11, 15, 10, 19, 21, 23, 25, 22, 18, 17, 13, 12, 11, 10]
55 [55, 53, 54, 50, 48, 52, 45, 51, 42, 40, 49, 37, 35, 47, 32, 46, 29, 27, 44, 23, 43, 20, 17, 41, 10, 11, 39, 4, 38, 8, 2, 36, 4, 2, 34, 10, 33, 11, 8, 31, 17, 30, 20, 23, 28, 27, 29, 32, 35, 37, 40, 42, 45, 48, 50, 53, 55, 54, 52, 51, 49, 47, 46, 44, 43, 41, 39, 38, 36, 34, 33, 31, 30, 28, 26, 24, 25, 21, 19, 16, 22, 9, 14, 6, 3, 18, 7, 5, 3, 15, 6, 9, 13, 5, 7, 12, 16, 14, 19, 21, 24, 26, 25, 22, 18, 15, 13, 1, 12, 1]
63 [63, 61, 62, 58, 56, 60, 53, 59, 50, 48, 57, 45, 43, 55, 40, 54, 37, 35, 52, 32, 51, 29, 27, 49, 23, 20, 47, 17, 46, 12, 9, 44, 10, 3, 42, 2, 41, 3, 2, 39, 9, 38, 12, 10, 36, 17, 20, 34, 23, 33, 27, 29, 32, 35, 37, 40, 43, 45, 48, 50, 53, 56, 58, 61, 63, 62, 60, 59, 57, 55, 54, 52, 51, 49, 47, 46, 44, 42, 41, 39, 38, 36, 34, 33, 31, 28, 30, 25, 26, 22, 19, 8, 18, 24, 11, 6, 4, 21, 5, 7, 8, 4, 6, 16, 5, 15, 11, 7, 13, 14, 19, 18, 22, 25, 28, 26, 31, 30, 24, 21, 16, 15, 13, 1, 14, 1]
64 [64, 62, 63, 59, 57, 61, 54, 60, 51, 49, 58, 46, 44, 56, 41, 55, 38, 36, 53, 33, 52, 29, 27, 50, 23, 20, 48, 17, 47, 12, 9, 45, 10, 3, 43, 2, 42, 3, 2, 40, 9, 39, 12, 10, 37, 17, 20, 35, 23, 34, 27, 29, 32, 33, 36, 38, 41, 44, 46, 49, 51, 54, 57, 59, 62, 64, 63, 61, 60, 58, 56, 55, 53, 52, 50, 48, 47, 45, 43, 42, 40, 39, 37, 35, 34, 32, 31, 28, 30, 25, 26, 22, 19, 8, 18, 24, 11, 6, 4, 21, 5, 7, 8, 4, 6, 16, 5, 15, 11, 7, 13, 14, 19, 18, 22, 25, 28, 26, 31, 30, 24, 21, 16, 15, 13, 1, 14, 1]
67 [67, 65, 66, 62, 60, 64, 57, 63, 54, 52, 61, 49, 47, 59, 44, 58, 41, 39, 56, 36, 55, 33, 30, 53, 26, 24, 51, 20, 50, 13, 11, 48, 12, 3, 46, 4, 45, 3, 7, 43, 4, 42, 11, 13, 40, 12, 7, 38, 20, 37, 24, 26, 35, 30, 34, 33, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 67, 66, 64, 63, 61, 59, 58, 56, 55, 53, 51, 50, 48, 46, 45, 43, 42, 40, 38, 37, 35, 34, 32, 29, 31, 28, 25, 23, 21, 27, 18, 9, 10, 2, 5, 22, 2, 8, 6, 19, 5, 9, 17, 10, 16, 6, 8, 14, 15, 18, 21, 23, 25, 29, 28, 32, 31, 27, 22, 19, 17, 16, 14, 1, 15, 1]
72 [72, 70, 71, 67, 65, 69, 62, 68, 59, 57, 66, 54, 52, 64, 49, 63, 46, 44, 61, 41, 60, 38, 36, 58, 33, 30, 56, 27, 55, 23, 20, 53, 17, 14, 51, 10, 50, 5, 3, 48, 4, 47, 3, 5, 45, 4, 10, 43, 14, 42, 17, 20, 40, 23, 39, 27, 30, 37, 33, 36, 38, 41, 44, 46, 49, 52, 54, 57, 59, 62, 65, 67, 70, 72, 71, 69, 68, 66, 64, 63, 61, 60, 58, 56, 55, 53, 51, 50, 48, 47, 45, 43, 42, 40, 39, 37, 35, 32, 34, 31, 28, 26, 24, 22, 29, 15, 13, 8, 6, 25, 7, 12, 9, 11, 21, 6, 8, 19, 7, 18, 13, 15, 9, 16, 12, 11, 22, 24, 26, 28, 32, 31, 35, 34, 29, 25, 21, 19, 18, 2, 16, 1, 2, 1]
75 [75, 73, 74, 70, 68, 72, 65, 71, 62, 60, 69, 57, 55, 67, 52, 66, 49, 47, 64, 44, 63, 41, 39, 61, 36, 33, 59, 30, 58, 26, 24, 56, 20, 17, 54, 14, 53, 5, 6, 51, 7, 50, 3, 5, 48, 6, 3, 46, 7, 45, 14, 17, 43, 20, 42, 24, 26, 40, 30, 33, 38, 36, 39, 41, 44, 47, 49, 52, 55, 57, 60, 62, 65, 68, 70, 73, 75, 74, 72, 71, 69, 67, 66, 64, 63, 61, 59, 58, 56, 54, 53, 51, 50, 48, 46, 45, 43, 42, 40, 38, 37, 35, 32, 29, 34, 28, 25, 23, 31, 12, 15, 9, 11, 27, 21, 4, 13, 10, 8, 22, 4, 9, 12, 19, 11, 18, 15, 8, 10, 16, 13, 23, 25, 29, 28, 32, 21, 35, 37, 34, 31, 27, 22, 19, 18, 2, 16, 1, 2, 1]
76 [76, 74, 75, 71, 69, 73, 66, 72, 63, 61, 70, 58, 56, 68, 53, 67, 50, 48, 65, 45, 64, 42, 40, 62, 36, 33, 60, 30, 59, 26, 24, 57, 20, 17, 55, 14, 54, 5, 6, 52, 7, 51, 3, 5, 49, 6, 3, 47, 7, 46, 14, 17, 44, 20, 43, 24, 26, 41, 30, 33, 39, 36, 38, 40, 42, 45, 48, 50, 53, 56, 58, 61, 63, 66, 69, 71, 74, 76, 75, 73, 72, 70, 68, 67, 65, 64, 62, 60, 59, 57, 55, 54, 52, 51, 49, 47, 46, 44, 43, 41, 39, 38, 37, 35, 32, 29, 34, 28, 25, 23, 31, 12, 15, 9, 11, 27, 21, 4, 13, 10, 8, 22, 4, 9, 12, 19, 11, 18, 15, 8, 10, 16, 13, 23, 25, 29, 28, 32, 21, 35, 37, 34, 31, 27, 22, 19, 18, 2, 16, 1, 2, 1]
83 [83, 81, 82, 78, 76, 80, 73, 79, 70, 68, 77, 65, 63, 75, 60, 74, 57, 55, 72, 52, 71, 49, 47, 69, 44, 42, 67, 39, 66, 36, 33, 64, 30, 27, 62, 23, 61, 17, 14, 59, 15, 58, 7, 4, 56, 5, 11, 54, 4, 53, 7, 5, 51, 14, 50, 17, 15, 48, 11, 23, 46, 27, 45, 30, 33, 43, 36, 39, 42, 44, 47, 49, 52, 55, 57, 60, 63, 65, 68, 70, 73, 76, 78, 81, 83, 82, 80, 79, 77, 75, 74, 72, 71, 69, 67, 66, 64, 62, 61, 59, 58, 56, 54, 53, 51, 50, 48, 46, 45, 43, 41, 38, 40, 37, 34, 32, 29, 26, 35, 25, 16, 13, 24, 31, 8, 6, 3, 28, 12, 9, 3, 10, 6, 8, 22, 13, 21, 16, 20, 9, 19, 12, 10, 18, 26, 25, 29, 24, 32, 34, 38, 37, 41, 40, 35, 31, 28, 22, 21, 20, 19, 2, 18, 1, 2, 1]
84 [84, 82, 83, 79, 77, 81, 74, 80, 71, 69, 78, 66, 64, 76, 61, 75, 58, 56, 73, 53, 72, 50, 48, 70, 45, 43, 68, 39, 67, 36, 33, 65, 30, 27, 63, 23, 62, 17, 14, 60, 15, 59, 7, 4, 57, 5, 11, 55, 4, 54, 7, 5, 52, 14, 51, 17, 15, 49, 11, 23, 47, 27, 46, 30, 33, 44, 36, 39, 42, 43, 45, 48, 50, 53, 56, 58, 61, 64, 66, 69, 71, 74, 77, 79, 82, 84, 83, 81, 80, 78, 76, 75, 73, 72, 70, 68, 67, 65, 63, 62, 60, 59, 57, 55, 54, 52, 51, 49, 47, 46, 44, 42, 41, 38, 40, 37, 34, 32, 29, 26, 35, 25, 16, 13, 24, 31, 8, 6, 3, 28, 12, 9, 3, 10, 6, 8, 22, 13, 21, 16, 20, 9, 19, 12, 10, 18, 26, 25, 29, 24, 32, 34, 38, 37, 41, 40, 35, 31, 28, 22, 21, 20, 19, 2, 18, 1, 2, 1]
87 [87, 85, 86, 82, 80, 84, 77, 83, 74, 72, 81, 69, 67, 79, 64, 78, 61, 59, 76, 56, 75, 53, 51, 73, 48, 46, 71, 43, 70, 39, 36, 68, 33, 30, 66, 27, 65, 23, 17, 63, 14, 62, 16, 7, 60, 12, 3, 58, 4, 57, 3, 7, 55, 4, 54, 14, 17, 52, 12, 16, 50, 23, 49, 27, 30, 47, 33, 36, 45, 39, 44, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 74, 77, 80, 82, 85, 87, 86, 84, 83, 81, 79, 78, 76, 75, 73, 71, 70, 68, 66, 65, 63, 62, 60, 58, 57, 55, 54, 52, 50, 49, 47, 45, 44, 42, 40, 41, 37, 35, 32, 38, 31, 28, 26, 24, 34, 8, 15, 9, 11, 6, 29, 13, 5, 10, 8, 25, 6, 9, 5, 22, 11, 21, 15, 20, 10, 13, 18, 19, 24, 26, 28, 32, 31, 35, 37, 40, 42, 41, 38, 34, 29, 25, 22, 21, 20, 18, 2, 19, 1, 2, 1]
95 [95, 93, 94, 90, 88, 92, 85, 91, 82, 80, 89, 77, 75, 87, 72, 86, 69, 67, 84, 64, 83, 61, 59, 81, 56, 54, 79, 51, 78, 48, 46, 76, 43, 40, 74, 36, 73, 33, 30, 71, 26, 70, 18, 15, 68, 17, 9, 66, 6, 65, 13, 14, 63, 4, 62, 6, 9, 60, 4, 15, 58, 18, 57, 17, 13, 55, 14, 26, 53, 30, 52, 33, 36, 50, 40, 49, 43, 46, 48, 51, 54, 56, 59, 61, 64, 67, 69, 72, 75, 77, 80, 82, 85, 88, 90, 93, 95, 94, 92, 91, 89, 87, 86, 84, 83, 81, 79, 78, 76, 74, 73, 71, 70, 68, 66, 65, 63, 62, 60, 58, 57, 55, 53, 52, 50, 49, 47, 45, 42, 39, 44, 38, 35, 32, 41, 31, 28, 34, 25, 37, 16, 12, 7, 11, 8, 3, 5, 10, 29, 3, 7, 27, 5, 8, 12, 11, 24, 16, 10, 23, 19, 20, 21, 22, 25, 28, 32, 31, 35, 39, 38, 42, 34, 45, 47, 44, 41, 37, 29, 27, 19, 24, 20, 23, 21, 2, 22, 1, 2, 1]
96 [96, 94, 95, 91, 89, 93, 86, 92, 83, 81, 90, 78, 76, 88, 73, 87, 70, 68, 85, 65, 84, 62, 60, 82, 57, 55, 80, 52, 79, 49, 46, 77, 43, 40, 75, 36, 74, 33, 30, 72, 26, 71, 18, 15, 69, 17, 9, 67, 6, 66, 13, 14, 64, 4, 63, 6, 9, 61, 4, 15, 59, 18, 58, 17, 13, 56, 14, 26, 54, 30, 53, 33, 36, 51, 40, 50, 43, 46, 48, 49, 52, 55, 57, 60, 62, 65, 68, 70, 73, 76, 78, 81, 83, 86, 89, 91, 94, 96, 95, 93, 92, 90, 88, 87, 85, 84, 82, 80, 79, 77, 75, 74, 72, 71, 69, 67, 66, 64, 63, 61, 59, 58, 56, 54, 53, 51, 50, 48, 47, 45, 42, 39, 44, 38, 35, 32, 41, 31, 28, 34, 25, 37, 16, 12, 7, 11, 8, 3, 5, 10, 29, 3, 7, 27, 5, 8, 12, 11, 24, 16, 10, 23, 19, 20, 21, 22, 25, 28, 32, 31, 35, 39, 38, 42, 34, 45, 47, 44, 41, 37, 29, 27, 19, 24, 20, 23, 21, 2, 22, 1, 2, 1]

For instructional purposes, here's the source code:

import java.util.*;

public class LangfordPairing {
    static void langford(int N) {
        BitSet bs = new BitSet();
        bs.set(N * 2);
        put(bs, N, new int[2 * N]);
    }
    static void put(BitSet bs, int n, int[] arr) {
        if (n == 0) {
            System.out.println(Arrays.toString(arr));
            System.exit(0); // one is enough!
        }
        for (int i = -1, L = bs.length() - n - 1;
                    (i = bs.nextClearBit(i + 1)) < L ;) {

            final int j = i + n + 1;
            if (!bs.get(j)) {
                arr[i] = n;
                arr[j] = n;
                bs.flip(i);
                bs.flip(j);
                put(bs, n - 1, arr);
                bs.flip(i);
                bs.flip(j);
            }
        }
    }
    public static void main(String[] args) {
        langford(87);
    }
}

Solutions for some N values are missing; they are known to require an abnormally large number of operations just to find the first solution by brute force.

Note that as mentioned, there are generally many solutions for any given N. For N = 7, there are 26 solutions:

[7, 3, 6, 2, 5, 3, 2, 4, 7, 6, 5, 1, 4, 1]
[7, 2, 6, 3, 2, 4, 5, 3, 7, 6, 4, 1, 5, 1]
[7, 2, 4, 6, 2, 3, 5, 4, 7, 3, 6, 1, 5, 1]
[7, 3, 1, 6, 1, 3, 4, 5, 7, 2, 6, 4, 2, 5]
[7, 1, 4, 1, 6, 3, 5, 4, 7, 3, 2, 6, 5, 2]
[7, 1, 3, 1, 6, 4, 3, 5, 7, 2, 4, 6, 2, 5]
[7, 4, 1, 5, 1, 6, 4, 3, 7, 5, 2, 3, 6, 2]
[7, 2, 4, 5, 2, 6, 3, 4, 7, 5, 3, 1, 6, 1]

[5, 7, 2, 6, 3, 2, 5, 4, 3, 7, 6, 1, 4, 1]
[3, 7, 4, 6, 3, 2, 5, 4, 2, 7, 6, 1, 5, 1]
[5, 7, 4, 1, 6, 1, 5, 4, 3, 7, 2, 6, 3, 2]
[5, 7, 2, 3, 6, 2, 5, 3, 4, 7, 1, 6, 1, 4]
[1, 7, 1, 2, 6, 4, 2, 5, 3, 7, 4, 6, 3, 5]
[5, 7, 1, 4, 1, 6, 5, 3, 4, 7, 2, 3, 6, 2]
[1, 7, 1, 2, 5, 6, 2, 3, 4, 7, 5, 3, 6, 4]
[2, 7, 4, 2, 3, 5, 6, 4, 3, 7, 1, 5, 1, 6]

[6, 2, 7, 4, 2, 3, 5, 6, 4, 3, 7, 1, 5, 1]
[2, 6, 7, 2, 1, 5, 1, 4, 6, 3, 7, 5, 4, 3]
[3, 6, 7, 1, 3, 1, 4, 5, 6, 2, 7, 4, 2, 5]
[5, 1, 7, 1, 6, 2, 5, 4, 2, 3, 7, 6, 4, 3]
[2, 3, 7, 2, 6, 3, 5, 1, 4, 1, 7, 6, 5, 4]
[4, 1, 7, 1, 6, 4, 2, 5, 3, 2, 7, 6, 3, 5]
[5, 2, 7, 3, 2, 6, 5, 3, 4, 1, 7, 1, 6, 4]
[3, 5, 7, 4, 3, 6, 2, 5, 4, 2, 7, 1, 6, 1]
[3, 5, 7, 2, 3, 6, 2, 5, 4, 1, 7, 1, 6, 4]
[2, 4, 7, 2, 3, 6, 4, 5, 3, 1, 7, 1, 6, 5]

Related links

• John Miller - Langford's Problem - Oddity for some N values
• OEIS A014552 - Number of solutions to Langford problem

Attachments

• Source code and output on ideone.com