The executioner and the hats (3 colores)

1. The first person may not be able to save himself, but he can leave useful information to others.
2. With three colors, this information can no longer be reduced to a simple opposition between two cases.
3. You look for a way to encode, in a single word, a global relationship between the colors you see.

Answer: 9 saved are guaranteed. Coding: red=0, blue=1, green=2 (module 3). Let $x_1,\dots,x_{10}$ be the real value of each hat. Person 10 (first to speak), who sees $x_1,\dots,x_9$, says:

$$ y_{10}\equiv-(x_1+\cdots+x_9)\pmod 3. $$ It may fail, but it leaves the equation fixed: $$

y_{10}+x_1+\cdots+x_9\equiv0\pmod3.
$$ Person 9 knows $y_{10}$, sees $x_1,\dots,x_8$ and solves for $x_9$:

$$ x_9\equiv-\big(y_{10}+x_1+\cdots+x_8\big)\pmod3. $$

He says it and he gets it right. Then person 8 already knows $x_9$ (heard), sees $x_1,\dots,x_7$ and solves for $x_8$; and so on until 1. All except the first are determined uniquely. Guarantee: 9 saved always; the first only has probability $1/3$ of being correct.