A problem on the relation between independence of pairs of random variables to conditional independence In my probability class, I was asked about this question concerning the possible relation between types of independence, reading as follows:

Let $ X,Y,Z $ be continuous (they have PDFs) and we are given two assumptions:

*

*Z and X are independent, and Z and Y are independent.


*Given Y, the variables X and Z are independent.
Now we are asked to prove or give a counterexample to 1 leads to 2 and vice versa.

The fact that X,Y,Z are continuous not discrete variables seems to make it more difficult.
I tried to show 1 leading 2 and the other direction simply using the definitions and such but got nothing so I am leaning to think that neither direction is correct, yet counterexamples elude my grasp. I am not sure they are correct or incorrect claims and I got no idea here, so I would truly appreciate the help on this. Thanks all helpers.
 A: Let $V_1,V_2,V_3$ be i.i.d. having your favourite distribution, for example $\mathcal N(0,1)$. Now define $X,Y,Z$ as follows: 
\begin{align}
X &= V_1+V_2\\
Z &= V_1+V_3 \\
Y &= V_1.
\end{align}
Obviously given $Y$ are $X$ and $Z$ independent, but neither pair $Z$ and $X$ nor pair $Z$ and  $Y$  are independent. 
A: Counterexample for (1) $\implies$ (2): Divide up the unit cube into eight congruent subcubes of side length $1/2$. Select four of these cubes: Subcube #1 has one vertex at $(x,y,z)=(1,0,0)$, subcube #2 has one vertex at $(0,0,1)$, subcube #3 has one vertex at $(0,1,0)$, and subcube #4 has one vertex at $(1,1,1)$. (Hope you can visualize this!). Now let $(X,Y,Z)$ be uniform over these four cubes. Then every pair of variables $(X,Y)$, $(X,Z)$, $(Y,Z)$ is uniform over the unit square (and hence independent), but conditional on one variable, the other two are not independent, because the cross-sections perpendicular to any axis are diagonally arranged squares.
If instead you select a different set of two subcubes--one with vertex at $(0,0,0)$ and the other with vertex at $(1,1,1)$--then you've got a counterexample to (2) $\implies$ (1): Conditional on one variable, the other two variables are independent, since the cross-sections perpendicular to any axis are squares (of side length $1/2$); but now every pair of variables is correlated, since their marginal distribution consists of two diagonally arranged squares.
