Интересное совпадение

На самом деле такая связь существует. Чтобы убедиться в этом, рассмотрим так называемый код Грэя. Инженер Ф. Грэй, работавший в 30-х годах в фирме AT&T Bell Laboratories, разработал его для исправления ошибок, возникающих при передаче информации электронными средствами. На рисунке внизу) приведены первые 22 двоичных числа натурального ряда и соответствующие коды Грэя.

При внимательном рассмотрении чисел, представленных в коде Грэя, можно заметить интересное совпадение. Коды Грэя, взятые в обратном порядке, совпадают с последовательностью конфигураций, ведущей к решению задачи «Китайские кольца»!

Таким образом, мы получили последовательность из 21 конфигурации для решения задачи с пятью кольцами, но для решения головоломки о башне с пятью дисками требуется 31 шаг. Можно предположить, что если эти задачи в каком-то смысле идентичны, то для их решения необходимо одно и то же число шагов. Несоответствие можно устранить, рассмотрев числа в коде Грэя, большие 21. Каждое из этих чисел представляет определенную конфигурацию в задаче с пятью кольцами, а последний код (соответствующий двоичному числу 11111), равный 10 000, представляет конфигурацию, в которой только одно последнее кольцо находится на петле. Это означает, что самое трудное исходное положение в задаче с кольцами такое, когда все кольца, кроме одного, ближайшего к рукоятке, сняты. При таком исходном положении число ходов, требующихся для решения задачи с л кольцами, равно 2я – 1 (точно тому же числу, что и для задачи с п дисками).

Очень интересную книгу, посвященную китайским кольцам, написал недавно С. Африа, профессор экономических и математических наук Университета г. Оттавы; она называется «Кольцо связанных колец» («The Ring of Linked Rings»). Именно ему принадлежит идея о связи между головоломкой «Китайские кольца» и понятием дуализма, символизируемого знаками «инь» и «ян». Хотя Африа выражает сомнение в китайском происхождении головоломки с кольцами, ему не удалось отыскать каких-либо определенных исторических сведений об игре, относящихся к периоду до 1550 г. В книге Африа описан также код Гро — предшественник кода Грэя, изобретенный в XIX в. французским математиком Л. Гро, опубликовавшим трактат об этой головоломке в 1872 г. Кстати, французы называют задачу с кольцами «бездельником» (le baguenodier), а англичане — «утомительными железками» (the tiring irons).

 

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