Prove that it is impossible to dissect a cube into finitely many cubes, no two of which are the same size

Answer and Step-by-step explanation:

Theoretically,

[Assuming a given Cube, C]

Suppose that dissection is possible.

Then, make the horizontal base to be a face of C. Divide the base into a perfect squared rectangle, R by the cube that are resting on it.

Each corner square of R has a smaller adjacent edge square, and the smallest edge square of R is adjacent to smaller squares that are not on the edge. Which means that the smallest square s1 in R is surrounded by larger, and thus higher, cubes on all 4 sides.

So the upper face of the cube on s1 is divided into a perfect squared square by the cubes that are resting on it.

In this dissection, let s2 be the smallest square. The sequence of squares s1, s2, s3, .... would be infinite and therefore have corresponding cubes that are in number.

This simply means dissection is not possible.

If it is possible for a 4-D hypercube to be perfectly hypercubed then it would have 'faces' that are perfect cubed cubes; but this is impossible.

Also, Cubes of higher dimensions are also impossible to be dissected.

Mathematically,

Assume that it is possible to represent a cube of length k, using smaller cubes and that the maximum number of smaller cubes needed to represent the larger cube to be is n.

In that case then, the total volume would be given in this expression (check the attachment) and it proved the impossibility of dissecting a cube.

the population of other specie of birds will decline or other specie of birds will find some alternate source of food to survive.

explanation:

according to the competitive exclusion principle, which is also known as gause's law, two specie living in the same niche and feeding on the same resource cannot co-exist stably.

as we have well studied the theory of natural selection which states that only those organisms survive which have better tendency to compete for food and adapt to the environmental conditions.

coming towards the question, what wi ll happen when one species outcompetes the other for the small fish , the other specie will not get enough food and will start to die or decline in population. eventually either other specie needs to find another alternative food source for itself or migrate at some other location where there is less competition otherwise it will just not exist anymore at that niche.

hope it !