Yes, I was going to suggest the covered boxes. and also possibly more boxes. It is possible that they throw it on the floor ONLY when they have trouble finding a clean spot. Can't say for sure, but it is a possibility. They do make liter boxes with the entrance on the top, but I am not sure Moose could quite make it into those. You may want to place the boxes inside yet another, say cardboard, box so it catches the litter.