ILL 1985 Problem 51

Let $f_1 = (a_1, a_2, \dots , a_n) , n > 2$ , be a sequence of integers. From $f_1$ one constructs a sequence $f_k$ of sequences as follows: if $f_k = (c_1, c_2, \dots, cn)$ , then $f_{k+1} = (c_{i_{1}}, c_{i_{2}}, c_{i_{3}} + 1, c_{i_{4}} + 1, . . . , c_{i_{n}} + 1)$ , where $(c_{i_{1}}, c_{i_{2}},\dots , c_{i_{n}})$ is a permutation of $(c_1, c_2, \dots, c_n)$ . Give a necessary and sufficient condition for $f_1$ under which it is possible for $f_k$ to be a constant sequence $(b_1, b_2,\dots , b_n), b_1 = b_2 =\cdots = b_n$ , for some $k.$