[BruceZ]

I got the cases wrong in the 3 color part. If the 10th guy sees 3 odds, the 9th guy will see 2 odds and 1 even (and he is the even color). If the 10th guy sees 1 odd, the 9th guy could also see 2 odds and 1 even OR 3 EVENS. Here is a picture of what the 9th guy sees:

o o e (3 odd, 9th is the e)
e e e (1 odd, 9th is any)
o o e (1 odd, 9th is one of the o's)

Here's the rule. If the 10th guy sees 1 odd color, he will say this odd color. If he sees all 3 colors odd, then he will say the color of the 9th guy's hat. So in all cases there is an odd number of whatever color he says. Then if the 9th guy sees ooe and hears the even color, he repeats this color which must be his since it must be odd. If the 9th guy sees eee, he also repeats the color the 10th guy says which must be his since it must be odd. If the 9th guy sees ooe and hears one of the odd colors, he says the other odd color from the one the 10th guy said. This color must be his. Now if the 9th guy and the 10th guy say different colors, everyone knows there is 1 odd, and it is the color the 10th guy says, so everyone knows what to do. If they say the same color, then if another player sees 3 evens, he knows the color they said must be odd, and that is his color. If they said the same color and another player sees 2 odds and an even, he knows there must have been 3 odds, and his color is the even one.

Here's the map again with what they say:

o o e (3 odd, 9th is the e, 10 and 9 both say same e)
e e e (1 odd, 9th is any, 10 and 9 both say same)
o o e (1 odd, 9th is one of the o's, 10 and 9 say different o's)