= Solution
For the <American put option>, compare exercise with continuation at time one. At $S_1=6$, both values are zero. At $S_1=4$, immediate exercise pays $1$, whereas the continuation value is the European value $4/9$, so exercise is optimal. At time zero, immediate exercise pays zero and continuation is
$$
R^{-1}\left(\frac{13}{16}\cdot0+\frac{3}{16}\cdot1\right)
=\boxed{\frac16}.
$$
The optimal policy is therefore to continue at time zero, exercise at time one if $S_1=4$, and otherwise continue to maturity. This nodewise maximum is the <Snell envelope> value.
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