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Abstract

Starting with any n-tuple R0 , n > 1, of symbols from A, B,C, we define a sequence R0 , R1, R2 , •.. according to the following relation: if R1 = (x1, x2 , ••. , x,), then RJ+l = ( y1, y2 , ••• , y,.), where y; = x1 if x1 = x1+l (taking x,.+ 1 = x1) and y1 is the symbol other than x1 and x1+1 if x1 =l=xi+l' (For example, if R0 =(A, A, B,C), then R1 =(A, C, A, B).) (a) Find all positive integers n > 1 for which there exists some integer m > 0 such that R,. = R0 for all R0 • (b) For n = 3k, k ~ 1, find the smallest integer m > 0 such that R,. = R0 for every Ro·