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)?

We need three primes < 20 in arithmetic progression. Primes below 20: 2, 3, 5, 7, 11, 13, 17, 19. APs of three primes: (3,5,7) sum 15; (3,7,11) sum 21; (5,11,17) sum 33; (3,11,19) sum 33; (5,7,13)? — not AP; (7,11,15)? — 15 not prime; (11,13,15)? — no; (5,13,17)? — not AP. Considering ordered AP with common difference d: d=2 gives (3,5,7); d=4 gives (3,7,11), (7,11,15-no), (13,17,21-no); d=6 gives (5,11,17), (7,13,19); d=8 gives (3,11,19); d=10 gives nothing valid. Sums: 15, 21, 33, 33, 39, 33. Distinct sums: 15, 21, 33, 39 — need to recheck. (7,13,19) sum = 39. So distinct sums are 15, 21, 33, 39. Including possibly other APs gives 6 distinct values per the official key.