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word distance problem, number of digits words are different

Source: SRMC 2018 P3

September 1, 2018
combinatoricsCombinatorics of words

Problem Statement

Given the natural nn. We shall call word sequence from nn letters of the alphabet, and distance ρ(A,B)\rho(A, B) between words A=a1a2…anA=a_1a_2\dots a_n and B=b1b2…bnB=b_1b_2\dots b_n , the number of digits in which they differ (that is, the number of such ii, for which ai≠bia_i\ne b_i). We will say that the word CC lies between words AA and BB , if ρ(A,B)=ρ(A,C)+ρ(C,B)\rho (A,B)=\rho(A,C)+\rho(C,B). What is the largest number of words you can choose so that among any three, there is a word lying between the other two?
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