Сознательное существование

Сможете ли вы полностью переместить башню из пяти карт прежде, чем наступит «конец света»? Согласно формуле 25 — 1, это можно сделать за 31 шаг (перемещение).

Как следует приступать к решению подобных задач? Почему некоторые люди, казалось бы, с легкостью решают головоломки, в то время как другим решение дается с трудом? В моем ответе на второй вопрос содержится и ответ на первый: я убежден, что каждый из нас пользуется математическим мышлением почти в любой момент своего сознательного существования. Как наши умозаключения относительно того, почему дядюшка Гарри не присутствовал на свадьбе, так и план размещения тюков и чемоданов в багажнике машины являются результатами логического вывода из определенных предпосылок. Процесс такого вывода может оказаться весьма изощренным, а это дает мне основания полагать, что почти каждый человек, способный проявлять такие интуитивные способности, может стать хорошим аналитиком. Задача заключается в том, чтобы довести интуитивные аналитические спбсобности до сознательного уровня.

Например, после того как человек «поиграл» некоторое время в задачу о башне из пяти дисков, он почти наверняка заметит, что время от времени возникает задача о башне с меньшим числом дисков. Довольно часто башня состоит из двух дисков, а иногда — из трех или даже четырех. Такие ситуации возникают сами собой, даже если чело-

век не имеет четкого плана действий, а просто играет с пирамидками.

Однако вскоре у играющего возникает ключевая идея: «Если можно построить пирамиду из двух или трех дисков (не говоря уже о пирамиде из четырех дисков), то почему бы не сделать пирамиду из пяти дисков?» Заметив определенные закономерности в процессе построения пирамидок меньшего размера, играющий рано или поздно приходит к той последовательности шагов, которая приводит к перемещению всей башни из пяти дисков на один из свободных стержней.

Аналогичная идея воплощается в математике и программировании в виде одного из распространенных методов решения задач. Если попытаться сформулировать принцип, лежащий в основе этого метода, то получится что- нибудь вроде: «Если можно решить задачу несколько меньшего размера, чем та, которую требуется решить, то, может быть, удастся воспользоваться решением, полученным для меньшей задачи, чтобы решить большую». Таким образом, мы приходим к понятию рекурсии — вложение в процедуру решения той же самой процедуры.

Идею рекурсии в применении к головоломке о башне сформулировать нетрудно. Если можно решить задачу для л — 1 дисков, то несомненно можно решить ее и для л дисков.

 

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