There's a few hundred sets of numbers that work, but none are the sequence 1,2,..,12 or so, due to needing balanced left and right weighings. To see this, write

There's a few hundred sets of numbers that work, but none are the sequence 1,2,..,12 or so, due to needing balanced left and right weighings. To see this, write the base 3 values for 1,2,..,12 as columns, and count the number of each digit in each row - you need equal numbers of 0 and 2 per row to make a valid weighing. Here I picked the above since it is a valid sequence, but fails for the reason elsewhere on this page - you also need to not have negative paired columns. And that too turns out to not be enough, so write a simple program and crunch to check each case that does work, and there's (I think) 304. I should stop poking at this as I have real work I should be doing :) If you really want to clean it up, you could then map the answer 3-vector of scrambled -1,0,1 values into another decoder matrix to map into values you like. Or use a look up table.