I plan to stop posting proposed answers until I've reviewed one several times!
However, here's a thought that might trigger a better thought: The number 17 is exactly the number you could cover if you made 4 tests with one-jar overlap. That is, if you tested
0-4, 4-8, 8-12, 12-16
This doesn't just jump up and tell me how to test them all in 8, nor does it point me in the direction of minimum-expectation test plan, but perhaps it will help someone.