Problem:
A die was thrown three times. The results of the throws are $X_1,X_2,X_3$. Let $A = X_1 + X_2$, $B = X_2 + X_3$ and $C = X_1 + X_2 + X_3$.
Find $p \Bbb Corr(A,C; B)$.
Note, $p \Bbb Corr(X, Y ;Z)$ means a partial correlation between $X$ and $Y$, "cleared" of the $Z$ effect.
Is it possible to use some formula? Using the formula for Pearson partial correlation I've obtained $p \Bbb Corr(A,C; B) = \sqrt{\frac{2}{3}}$ but is it right?
I believe $\sqrt{\frac{2}{3}} \approx 0.8165$ is correct
As a hand-waving check, it seems fitting $A$ against $B$ gives a coefficient of $\frac12$ to $B$, while fitting $C$ against $B$ gives a coefficient of $1$ to $B$.
So you are looking for the correlation between $A-\frac12B$ and $C-B$, i.e. between $X_1+\frac12X_2-\frac12X_3$ and $X_1$.
We do not need to know the variance for a single die is $\sigma^2 =\frac{35}{12}$, as the variance of the first term is $\sigma^2+\frac1{2^2}\sigma^2+\frac1{2^2}\sigma^2=\frac32 \sigma^2$ while the variance of the second is $\sigma^2$ and the covariance between these is the varaince of $X_1$ so is also $\sigma^2$, making the correlation $\dfrac{\sigma^2}{\sqrt{\frac32 \sigma^2\, \sigma^2}} = \sqrt{\dfrac{2}{3}}$