Вопросы Теги

Как получить определенную последовательность, подобную этой?

Есть 100 чисел:

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

Как получить последовательность, в которой один номер находится между двумя 1s, два числа между двумя 2s, три числа между двумя 3s, .. и пятьдесят чисел между двумя 50-ми, используя сто чисел?

У кого-нибудь есть идея лучше, чем грубая сила? Или докажите, что нет решения, когда n = 50.

11
задан ЯegDwight 8 September 2012 в 00:15
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5 ответов

Есть только решение этой проблемы для пар чисел от 1 до n, где n = 4m или n = 4m-1 для любого положительного целого числа m.

Обновление:

Для любого решения пары нечетных чисел должны занимать две нечетные или две четные позиции. Пары четных чисел должны занимать по одной каждой. Когда имеется нечетное количество нечетных пар (например, 1 1 2 2 3 3 4 4 5 5 - 3 нечетных пары) решения нет. Невозможно разместить первое число каждой пары, так как это не приведет к конфликту, когда вы попытаетесь разместить второе.

См. http://en.wikipedia.org/wiki/Langford_pairing

Другое обновление:

Мой ответ был в основном от Кнута. Я все обдумал и самостоятельно придумал следующее.

Для любой последовательности {1 1 2 2 ... n n} существует, скажем, m нечетных пар, n-m четных пар и 2n позиций, в которые их можно разместить (т.е. n позиций каждой четности).

Если вы сначала разместите четные пары, то вы используете n-m четных позиций и n-m нечетных позиций, таким образом, у вас останется m позиций каждой четности, в которые можно разместить нечетные пары.

Нечетные пары должны быть размещены на позициях с одинаковой четностью. Если m четное, проблем нет, потому что половина пар будет размещена в нечетных позициях, а половина - в четных.

Если m нечетно, вы можете разместить только m-1 нечетных пар, после чего у вас останется разместить одну нечетную пару и по одной позиции каждой четности.Поскольку для нечетной пары требуются позиции с одинаковой четностью, нет решения, когда m (количество нечетных пар) нечетно.

6
ответ дан 3 December 2019 в 04:30
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Я почти уверен, что видел это в книге Кнута «Искусство компьютерного программирования». Я посмотрю, когда вернусь домой сегодня вечером.

1
ответ дан 3 December 2019 в 04:30
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Я думаю, что хорошей эвристикой будет начать с самых больших чисел. Тем не менее, вам все равно понадобится возврат.

Проблема заключается в том, что наличие решения для n-1 обычно мало помогает найти решение для n.

0
ответ дан 3 December 2019 в 04:30
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Всегда ли это возможно? Я вижу, что закономерность вероятна, а не определена. Это не работает для единиц, единиц и двоек.

Если это вероятно, лучше всего было бы начать с грубой силы и продолжать устранять неправильные варианты, когда вы с ними сталкиваетесь.

0
ответ дан 3 December 2019 в 04:30
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Проблема: Langford Pairing

Проблема известна как Langford Pairing. Из Википедии:

Эти последовательности названы в честь К. Дадли Лэнгфорда, который поставил проблему их построения в 1958 году. Как описывает Кнут1, проблема перечисления ВСЕХ пар Лэнгфорда для заданного N может быть решена как экземпляр проблемы точного покрытия (одна из 21 NP-полной проблемы Карпа), но для больших N количество решений может быть вычислено более эффективно алгебраическими методами.

1: Искусство компьютерного программирования, IV, Фасцикл 0: Введение в комбинаторные алгоритмы и булевы функции

Стоит отметить, что для N = 50 решения не существует. Решения существуют только для N = 4k или N = 4k - 1. Это доказано в работе Роя А. Дэвиса (On Langford's Problem II, Math. Gaz. 43, 253-255, 1959), где также приводится схема построения единственного решения для любого выполнимого N.

Ссылки по теме

Перебор в Java

Вот несколько решений, которые смогла найти моя быстрая и грязная программа перебора для N < 100. Почти все они были найдены менее чем за секунду. 

 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]

В учебных целях вот исходный код:

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);
    }
}

Решения для некоторых значений N отсутствуют; известно, что они требуют аномально большого количества операций, чтобы найти первое решение методом перебора.

Заметим, что, как уже говорилось, для любого заданного N обычно существует множество решений. Для N = 7 существует 26 решений:

[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]

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ответ дан 3 December 2019 в 04:30
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