Consider a square of 4 units by 4 units centered it the orig

Consider a square of 4 units by 4 units centered ;it the origin. The two coordinate axes divide this square into 4 compartments, with one in each quadrant. We can now draw a unit circle in each of the 4 compartments. Finally, we can draw a small circle at the origin tangent to the 4 compartment circles. We can do a similar construction in the three dimensional space. Consider a box of size 4 units by 4 units by 4 units. The three coordinate planes will divide the box into 8 compartments. We can then draw a unit sphere in each of these 8 compartments, and finally draw a small sphere centered at the origin tangent to these 8 compartment spheres (figure 2). We now generalize this construction to the n-dimensional space. Consider a box centered at the origin w hose size is 4 unit long in each dimension. This box consists of all the points (x_1, x-2,...,x_n) in the n-dimensional space, where - 2

Solution

consider the diagonal of smaller n dimensional cube(of side length 2). we have n dimensional larger sphere inscribed in it with radius 1. length of diagonal = 2 \\sqrtn

half the length of diagonal - radius of larger sphere gives the radius of smaller sphere.

it is

\\sqrt n - 1 .

We see that it keeps on increasing

for n = 9 it is 2

n= 16 it is 3

even greater than the radius of larger sphere.

there is something

our assumptin is not right that balls will touch each other