Posted for "John Barrett" <2thman@cablespeed.com>:

Let me try that again. If you are lucky just weigh one. If it’s the heavy
one, then you’ve done it in one measurement – that’s the answer to the
question.

Assuming you are unlucky and need to find the quickest path of eliminating
the 623 gram cylinders, Then I think the number of steps is five. It is
conditional, but either path requires five measurements to get to one
remaining piston. You would pick groups of cylinders to weigh in one of the
following two sequences:

1. 13,6,3,2,1
2. 13,7,4,2,1

Obviously, you would select each group by halving or halving minus one
piston (even or odd group?) from each succeeding “heavy”. Just divide the
weight of each sample by 623. If there is one gram left over, then the next
halving should be taken randomly from that group. Otherwise, grab from the
pistons you randomly did not select to do the most current weighing.