Risk neutral probability for each of 3 states
I cannot really follow your formulas, what is the logic behind them?

Seems to me there is no way to divine three state risk-neutral probabilities from 1 financial instrument's prices. The equation
$$
p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 = 180
$$ 
is underdetermined, there are infinitely many solutions to it. Now if you solved the equation system
$$
\begin{align*}
p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 &amp; = 180 \\
p_1 94 + p_2 134 + (1 - p_1 - p_2) 54 &amp; = 100
\end{align*}
$$
there you have two unknowns and two equations, so usually this would admit exactly one solution.

P.s.: The numbers in my example are not corrected for intertemporal differences, there has been no discounting.