For the third puzzle the hint of infinite potential is the key. By moving the 2nd pawn from the right all the way to the other side, you can exchange it for a knight. Use both knights to occupy the 1st and 2nd spots from the left in front of the king, the king is now in checkmate.
The fourth puzzle just is. You will need a queen covering the diagonal black spaces the king can move to, while using the white bishop to force it to move. If you put a white knight in between the two pawns and the black knight, then you have checkmate and another trainer battle to look forward to.
A telecommunicator is located in entrance.
AH, you're challenging Radomus. Radomus is a Doubles leader whose strategy focuses around Trick Room. His team is bulky and hits hard, and if he sets up, then it's going to be rough going. But if you can stop him from using Trick Room, then he might as well be a sitting duck.
An another note, neither side is alone in this battle. The chess pieces may come into play too. Radomus can move them with his psychic attacks, dealing extra damage, but you might be able to find ways to move them too. Keep in mind that like any good chess game, Pokemon with high intellect or strategy-based attacks will do well here.
Cain is battled on a Chess Board in a Single Battle format and he has two Moomoo Milks.