Большие шансы на ошибку

Начиная с этого шага, играющий при выборе каждого следующего хода имеет все большие шансы совершить ошибку. Но даже в том случае, если все перемещения будут выбраны правильно, потребуется 2" – 1 шага, чтобы переместить башню из дисков по одному на каждом шаге, на другой стержень (число шагов мы вычислим ниже). Чтобы решить эту головоломку даже при сравнительно небольшом числе дисков, требуется поразительно много времени. Это прекрасно проиллюстрировано в легенде, которую мы процитируем по классической книге Болла о головоломках (W.R. Ball «Mathematical Recreations and Essays»).

«В великом храме Бенареса. под куполом, находящимся точно в центре мира, покоится бронзовая пластина с тремя алмазными стержнями, каждый длиной в один локоть и толщиной как тело пчелы. На один из этих стержней бог при сотворении мира положил шестьдесят четыре диска из чистого золота, так что наибольший диск лежит на бронзовой пластине, а остальные образуют пирамиду, самый маленький диск находится на вершине пирамиды. Это башня Брамы. Работая день и ночь, жрецы переносят диски с одного стержня на другой, следуя мудрым и незыблемым законам Брамы, согласно которым диски можно перекладывать только по одному и каждый диск можно класть на стержень только в том случае, если под ним не окажется диск меньшего размера. Когда все шестьдесят четыре диска будут таким образом перенесены, и башня, и храмы, и жрецы-брамины превратятся в прах, и наступит конец света».

То, что мир до сих пор не исчез, свидетельствует о чрезвычайно большом времени, требующемся для решения задачи: даже если жрецы будут перекладывать по одному диску в секунду, то потребуется более 500 млрд. лет, чтобы переместить всю башню из 64 дисков!

А теперь (без всякого риска для Вселенной) читатель может ближе познакомиться с этой задачей, взяв пять игральных карт, например последовательно от туза до пятерки пик, и условно выделив три места на столе, куда можно было бы класть карты. Сложите эти карты колодой в одном из трех мест по порядку, так чтобы туз находился сверху. Попытайтесь решить головоломку о башне с пятью дисками, перемещая карты по одной с места на место, но так, чтобы ни одна карта не легла на карту меньшего достоинства (в данном случае считается, что двойка «старше» туза).

 

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