Will the box float? If you submerse an object in water it will float if its density is less than that of water. This is because pressure increases linearly in water as you go deeper, so the bottom of the object experiences more pressure. If you sum up all the forces you will get an upwards force proportional to the volume of the object, regardless of what's in it. Like this box:

But what now if we put this box on the bottom of the ocean floor. There will be no water at the bottom anymore to exert pressure. The pressure forces will only point downward so you would expect a large downwards force. There will also be a normal force equal to the weight of the box, so in total you would have zero net force (I think). So is it possible to stick boxes to the bottom of the ocean floor like this? Because my intuition tells me it would still float.

 A: It won't float if you place them on top of one another perfectly, which may seem counterintuitive. Normally, you would have a situation where you would have a slight gap between the 2 boxes due to either different materials used or simply because of the formation of oxide on the surface of one material. Then, the water molecules can do their job. However, if you manage to remove all the impurities on the surface, the box will in fact stick. In essence, you can't distinguish that the two boxes are separate and they will appear as 1 box. The same thing happens in case of friction when you try to smoothen a material to reduce friction. After enough smoothening, the 2 materials just get joined together because the molecules can't differentiate that the 2 surfaces are separate. Also, in your example, the slight gap I mentioned previously would also be between the ocean floor and the box. So, even your original box will start to float. 
A: You have a similar situation when you lay a box on a table in air. You might expect the box to be firmly held down by a force of 14.7 lb on each square inch of the top. Usually you do not see this because there is enough air under the box that you do get very much the same upward force. 
If you take two extremely flat pieces of metal and carefully put them together so there is no air between them,  you do get this force. It can be hard to get them back apart. 
Machinists take advantage of this to make very precise measurements. They make or buy an extremely flat block with extremely parallel faces that are $1.000000$ inches apart. And another with faces $0.500000$ inches apart. And as many more blocks as they can afford. As little  as $0.000001$ inch of air between the blocks would spoil their measurements. If the blocks stick together, they know there is less than that much
Google gauge block
