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alphabet of language BAU consists of letters B, A, and U

Source: 2004 Estonia National Olympiad Final Round grade 11 p5

March 25, 2020
alphabetcombinatorics

Problem Statement

The alphabet of language BAUBAU consists of letters B,AB, A, and UU. Independently of the choice of the BAUBAU word of length n from which to start, one can construct all the BAUBAU words with length n using iteratively the following rules: (1) invert the order of the letters in the word; (2) replace two consecutive letters: BA→UU,AU→BB,UB→AA,UU→BA,BB→AUBA \to UU, AU \to BB, UB \to AA, UU \to BA, BB \to AU or AA→UBAA \to UB. Given that BBAUABAUUABAUUUABAUUUUABBBBAUABAUUABAUUUABAUUUUABB is a BAUBAU word, does BAUBAU have a) the word BUABUABUABUABAUBAUBAUBAUBBUABUABUABUABAUBAUBAUBAUB ? b) the word ABUABUABUABUAUBAUBAUBAUBAABUABUABUABUAUBAUBAUBAUBA ?
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