We have boxes and we are picking up a wood and put in between our hand and the box. We start pushing wood with 20N force. This force Is transferred from wood to the box and box return 20N to wood (Newton’s third law). So we have 20N from left to right and 20N from right to left that have an effect on box1. So why the wood (box1) start moving? (You can use any force instead 20N.) (Please check image).
You can't say that the contact force has a magnitude of 20 N.
To find the contact force, firstly find the acceleration of the system, which will be $\frac{net\: external\: force}{net\: mass}$. After this, you can draw the individual f.b.ds to find the contact force. This I leave to you.
The major error was directly stating that the first block exerts 20N on the second block
Your free body diagram already has the answer. There are two 20N vectors pointing right and one pointing left. $\sum F = ma$. If you follow the chain of forces back through your hand, arm, body, etc, you'll keep finding one extra instance of 20N right until you get to the planet you're standing on, which balances out with one extra instance of 20N left.
