Touring a chess board with multiple pieces [Python]

Question

I have been trying to solve this for 2 days straight and I just can't find a valid algorithm. Given a Chess board, and a number of pieces, I have to check if said board can be toured by the pieces, with the condition that each piece can only visit an square once. I know it's some kind of multiple backtracking, but I can't get it to work. (I only have been able to implement a general knight's tour for individual pieces)

tablero is a class of the board, that holds a name, a list of pieces, a list with prohibited positions, a list with the free positions, and a tuple with the dimensions of the board.

ficha is the class of a piece, it holds a name (nombre), a tuple with its position (posicion), a list with its valid movements (movimientos) (For example, a pawn's list would be [ [0,1] ], meaning it can only move forward 1)

Any insight is welcome.

Here are the classes (Feel free to add/remove any method).

def legal(pos,dimensiones):
    if pos[0] >= 0 and pos[0] < dimensiones[0] and pos[1] >= 0 and pos[1] < dimensiones[0]:
        return True
    else:
        return False


class board:

    def __init__(self,name,pieces,dimention,prohibited_positions):
        self.name = name
        self.pieces = pieces
        self.dimention = dimention
        self.prohibited_positions = prohibited_positions
        self.free_positions = []
        for x in range(dimention[0]):
            for y in range(dimention[1]):
                self.free_positions.append([x,y])
        for x,y in self.prohibited_positions:
            if [x,y] in self.free_positions:
                self.free_positions.remove([x,y])
        for piece in self.pieces:
            if self.piece.position in self.free_positions:
                self.free_positions.remove(piece.position)


    def append(self,piece):
        pos = piece.position
        if pos in self.free_positions:
            self.pieces.append(piece)
            self.free_positions.remove(pos)



class piece:

    def __init__(self,name,position,move_offsets):  
        self.name=name
        self.position=position
        self.move_offsets=move_offsets
        self.possible_movements=move_offsets

    def setPos(self,pos):
        self.position=pos

    def ValidMovements(self,dim,free_positions,prohibited_positions):
        aux = []
        for i in self.possible_movements:
            newX = self.position[0] + i[0]
            newY = self.position[1] + i[1]
            newPos = [newX,newY]
            if legal(newPos,dim):
            aux.append(newPos)
        for i in list(aux):
            if i not in free_positions:
                aux.remove(i)