I can solve a Rubik's Cube underwater in a swimming pool but my problem is that I can't stay at the bottom of the pool because of the oxygen in my lungs. How can I stay in the bottom without going up to the surface?
Paparazzi's suggestion from comments, to dive with a weight (not bound to your body, for safety reasons -- the weight should drop away when you straighten to push off for the surface, at the least) that will overcome your natural buoyancy. You'll need to experiment to see how much weight is needed.
A milk jug or similar, partially filled with sand and then topped up with water to eliminate air, is a good adjustable weight (more sand makes it heavier under water, but the water portion is weightless when submerged). You can also tie a rope on the handle with a float on the other end, to make it easy to retrieve the weight from the bottom of the pool. Most "normal" people will require between 5 and 10 kg of sand, perhaps less if the weight is iron (like a weight plate from a barbell set) or lead (like scuba divers' weights).
Very Important!! Never do this without a "buddy", someone who can push the weight off you and assist you to the surface if you go over your allotted solving time -- if you lose consciousness underwater, alone, you will die.
NOTE: This all depends on how long you can hold your breath.
When a rocket launches, the force from the bottom of the rocket, in kids words, "fire", pushes the rocket up in simple means.
However, if the pressure was coming from the TOP of the rocket, the rocket will go down instead of up.
Now, take a very deep breath and jump into the pool. As soon as your nose goes underwater start breathing out. This is similar to the example I gave you earlier, it's like the pressure making the rocket fly, but instead the air is pushing you down. Now, you'll start going back up. When that happens start moving your arms in a way that it looks like you're "pushing the water. You can stay down for a while, but this all matters on how long you can hold your breath.
My friend texted my and told my a solution that worked for him. The answer for him was to have dumbbells on his thighs while being underwater. He even showed my a video where it prove that it can be done.
It can be watched here https://www.facebook.com/671704706/videos/vb.671704706/10154639991754707/?type=3&theater