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Suppose I have a cube with one open side (with a volume of let's say $1\ m^3$) for the sake of simplicity; the problem is scale invariant) made from a material that makes the cube just float in water if half the cubes volume is filled with water. Now I put this cube on a table, half filled with water. Then I put another cube in it (all cubes are made from the same stuff). This cube has half the volume of the bigger one, and I fill it half with water. In the last cube, I put a cube that has half the volume of the one before and again I fill it half with water. And so on. Ad infinitum.

What will happen to the water level with respect to the level where the first cube was half filled? Do things overflow, is the level getting down again after the first cube I put in (wich clearly makes the level rise) or what?

Deschele Schilder
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  • "material that makes the cube just flow in water if half the cubes volume is filled with water." I have no idea what this means. Do you mean float, instead of flow? And if so, how is the cube floating if the water is inside it? – Pockets Apr 26 '16 at 18:12
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    Hint: Try to figure out the total amount of water in the biggest cube. – flawr Apr 26 '16 at 18:12
  • You talk as if the material alone determines the cube's weight. It's actually the material and the thickness of its walls together. (I gather that it's a hollow cube with walls surrounding a cavity.) You say that all cubes are made from the same stuff -- I suspect that would you want to imply by that is that they all just float in water when they're half filled with water? If they're all made of the same material, that would require their wall thickness to shrink in proportion to their size. – joriki Apr 26 '16 at 19:48
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    As the problem stands, it's not well-defined what will happen. When you put the second cube into the first cube, it makes the water rise to the top of the first cube, since the second cube, being half filled with water, just floats, and thus makes the water rise until the cubes' open faces coincide, with half the first cube filled by water and the other by the second cube. If you now add the third cube, it will push the second cube down; the water in the first cube can spill both to the outside and into the first cube, and we can't decide mathematically which of the two it will do (or both). – joriki Apr 26 '16 at 20:00
  • The material of each cube is the same and has the right thickness to let it float in water (the open side has the same level as the water) if you put half the cube´s volume water in it. Wich means (scale invariance) that the walls are getting thinner (3 times) every time I put the next cube in.But this talk about the material is not my question. Joriki´s second answer and flawr´s answer are more to the point.Joriki and flawr make it clear that when I put the half filled, second cube in the first one, the openings coincide and the water level reaches 1 (m).After the next it is unclear. Thanks! – Deschele Schilder Apr 26 '16 at 20:49
  • Just one more question. Has the indeterminance after puting the third cube in a connection with chaos theory? – Deschele Schilder Apr 26 '16 at 20:57
  • Again just one more question. The ratio of volumes between one cube and the one before I took 0.5. There is a ratio (somewhere between 0 and 0.5) for wich the limit gives a situation in wich all the water júst stays in the cubes. Can you calculate that value? – Deschele Schilder Apr 27 '16 at 07:31
  • If the ratio is 0.5, like in the example doesn´t the second cube sink after you put the third cube in, so the water level in the first cube goes down? – Deschele Schilder Apr 27 '16 at 07:58

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