10 There are six thin sticks, among which the two longer ones are $\sqrt{3} a$ and $\sqrt{2} a$, and the other four are all $a$. Using them to form a triangular pyramid, the cosine value of the angle between the two longer edges is
(10) As shown in the figure, at first glance, two diagrams can be made, as shown in Figure 1 and Figure 2. However, if we slightly consider the existence of the two diagrams, we immediately find that Figure 1 does not exist. By taking the midpoint $E$ of $AC$ and connecting $BE$ and $ED$, we get $BE = ED = \frac{\sqrt{2}}{2} a$, thus
$$
BE + ED = \sqrt{2} a < \sqrt{3} a = BD,
$$

which contradicts the triangle inequality that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, Figure 1 should be excluded. In Figure 2, the cosine of the angle between the two longer edges is $\cos \angle BDC = \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{6}}{3}$.