Showing the pairwise XORed random variables are independent
If two binary RVs are uniform, they are independent if and only if the probability that they are equal is the same as the probability that they are unequal (so knowing one does not help in guessing the other).
$$
P[X_1 \oplus X_2=X_2\oplus X_3]=P[X_1=X_3] =P[X_1\oplus X_3=0]
$$
which is clearly $1/2$ since $X_1,X_3$ are unbiased and independent.