I know that the number of degrees of freedom of a system composed of M particles subject to j constraints is: $$ DOF = 3M - j$$. Now my book mentions that the number of $DOF$ of a rigid body with more than three mass points is always 6. However, why is that the case? What is the proof of this statement? For example in the picture below, I would think that the number of constraints $j$ is: $$j = 3 lenghts + 2 angles $$ and therefore $DOF = 4*3 - 5 = 7$. Where am I wrong?
There is one more angle in that rotation around the central stick is not allowed. If the four masses are in a plane they must remain in a plane. If the right three are in the plane of the paper and the left one is in the vertical plane through the middle stick it must remain that way. For a rigid body there are three angular degrees of freedom. In your drawing we can start from the right hand mass. You have two angles to position the next mass, then one rotation around the line between the first two masses to locate the third. After that the location of all the points is fixed.