So let me get this straight. 10 gnomes in a line, wearing hats that are 1 of 3 possible colors. I have to guess each gnome's hat (10 guesses). The gnomes can talk to each other, but not me, and they can arrange themselves in the order of their choosing? Well you can save AT LEAST 5 every time. I say 6 because think of it this way, let's say the colors are red, blue, and green:
rrrbbbbggg
Well, let's say you guess blue first, well now we have 9 left:
rrbbbbggg
Well then you guess green:
rbbbbggg
Well now you know the gnomes put the reds first, so you guess red, that's 1 saved. Now you guess red again, because you don't know how many of the 10 gnomes have red hats, well now one blue is dead, so we have:
(r)bbbggg
Well you know it's not red, so now if you guess green, you get:
(r)bbggg
So we know it's blue now, so you guess blue, and save a gnome, guess blue, save another gnome, and guess blue again, because like before, we don't know how many are blue, so now we have:
(r)(b)(b)gg
Well now all that's left is green, so we guess green, and green again:
(r)(b)(b)(g)(g)
And thus, you've saved half. Now if you adjust how many of each color there are, using the same logic (assuming the gnomes are smart enough to group themselves together like that), you'll save AT LEAST 5 each time. You could save more if you didn't happen to guess wrong the first time each time.
Two flaws with this riddle though. 1) You never said I couldn't see the hats, thus I could just look at the hats and save all 10. 2) You didn't say the gnomes couldn't talk to me, the gnomes could just tell me their hat color each time, and thus I could save all 10.
