Tiled Pool

Let the square tile have side 'x'. Also let the rectangular pool has length 'l', breadth 'b' and depth 'h'. Thus from the given statement we get two equations,

l b=231 { x }^{ 2 }

2 h (l+b)=1024*{ x }^{ 2 }

the first equations gives factors as 3,7,11. Thus three possible solutions are there: (33,7), (77,3) and (21,11). (21,11) is viable as (l+b) can't be a multiple of 10, from the latter equation.

Thus we get, h=16x. Hence number of tiles in line is 16.

Take the square tiles for unit of measurement. The area of the base of the pool is 231 units. Now prime factorise 231, 231 = 3 11 7 Now, there are three cases of l*b:-

3*77..................case 1

21*11................case 2

33*7...................case 3

Now, we individually look at the perimeter made by all the cases.

In case 1, p = 160

In case 2, p = 64

In case 3, p = 80

Now, the area of 4 walls = perimeter * height = 1024. If we apply some logic, if p*h = 1024, h can't be in decimals [given in question]. So, for p=160 and p= 80, value of h doesn't fits the question. So, only for p = 64, values of h, i.e 1024/64 = 16 fits in the question's conditions. Hence, height, that is the number of tiles that line the depth of the pool is 16 units or 16 tiles.

By trial and error method. The possible solutions for no. of tiles along length and breath were (3,77), (7,33) & (11,21). The no. of tiles along the height = 16.