= Solution
{c}
Conditional on $\mathcal F_n$, the increment is $-X_n/2$ with probability $1-X_n$, and $(1-X_n)/2$ with probability $X_n$. Therefore
$$
\mathbb E((X_{n+1}-X_n)^2\mid\mathcal F_n)
=\frac{(1-X_n)X_n^2+X_n(1-X_n)^2}{4}
=\frac{X_n(1-X_n)}4.
$$
Taking expectations using the <tower property> yields
$$
\boxed{\mathbb E(X_{n+1}-X_n)^2=\tfrac14\mathbb E[X_n(1-X_n)].}
$$
This is the conditional variance that drives the <Bernoulli polarization martingale> toward its endpoints.
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