new puzzle challenge

[Don Shepherd]

I was browsing at my local Barnes & Noble Bookstore today and came across this number puzzle that struck my fancy:

38 puzzle

It consists of a little wooden frame containing 19 removable hexagonal pieces, numbered 1 through 19. There are a total of 15 rows in all directions, each made up of 3, 4, or 5 pieces (5 parallel rows starting with 9, 11, 18; 5 parallel rows starting with 9, 14, 15; and 5 parallel rows starting with 18, 17, 3). To goal is to have the sum of each of the 15 rows be 38. The picture shows one solution, but the website of the puzzle maker says there is more than one solution.

I'd like to find all possible solutions. A brute-force approach would require examining 19! possible configurations, a very large number (almost as many configurations as an Enigma machine). I'm sure Alan Turing could figure this out if he were alive today, but he is not. Who among us will pick up the gauntlet?

I expect a one line RPL solution, as usual! I would love to see a BASIC solution.