MathDB
Reversing sequence of cards by shuffling

Source: Japanese MO Finals 2000

February 10, 2011
functionmodular arithmeticcombinatorics proposedcombinatorics

Problem Statement

Let 3n3n cards, denoted by distinct letters a1,a2,…,a3na_1,a_2,\ldots ,a_{3n}, be put in line in this order from left to right. After each shuffle, the sequence a1,a2,…,a3na_1,a_2,\ldots ,a_{3n} is replaced by the sequence a3,a6,…,a3n,a2,a5,…,a3n−1,a1,a4,…,a3n−2a_3,a_6,\ldots ,a_{3n},a_2,a_5,\ldots ,a_{3n-1},a_1,a_4,\ldots ,a_{3n-2}. Is it possible to replace the sequence of cards 1,2,…,1921,2,\ldots ,192 by the reverse sequence 192,191,…,1192,191,\ldots ,1 by a finite number of shuffles?
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