USAMO 1990 Problem 3

Show that for any odd positive integer we can always divide the set $\{\mathrm{n}, \mathrm{n}+1, \mathrm{n}+2, \ldots, \mathrm{n}+32\}$ into two parts, one with 14 numbers and one with 19 , so that the numbers in each part can be arranged in a circle, with each number relatively prime to its two neighbours. For example, for $\mathrm{n}=1$, arranging the numbers as $1,2,3, \ldots$, 14 and $15,16,17, \ldots, 33$, does not work, because 15 and 33 are not relatively prime.