The puzzle is 500 pieces and the border used 75 pieces so 500-75 = <<500-75=425>>425 pieces left
Joe placed three times as many puzzle pieces as Trevor who placed 105 so Joe placed 3*105=315 piece
Together Joe and Trevor placed 315+105 pieces for a total of <<315+105=420>>420 pieces
With the border down that left 425 pieces left and they put down 420 pieces so that meant 425-420 = <<425-420=5>>5 pieces were missing
#### 5