Three prime numbers p, q and r, each less than 20, are such that p - q = q - r. How many distinct possible values can we get for (p + q + r)?

p, q, r form an arithmetic progression, so p + q + r = 3q. Equal primes (p = q = r) give 3q for all 8 primes below 20, and genuine progressions such as 3, 5, 7 add no new q values beyond these, so there are more than 6 values. (Answer as per the official UPSC answer key.)