Rectangular Ground with Square tiles

Question

A rectangular ground is to be filled with square tiles of unit area. The border of the ground should be of red coloured tiles and inside white. If the number of red and white tiles are the same, find the minimum dimensions of the ground.

Answer 1

The diophantine equation is $ab = 2(a-2)(b-2)$. To solve it, first suppose both of the numbers are at least 7. Since $\left(\frac{7}{5}\right)^2 < 2$, in this case there are no solutions. Thus, at least one of the numbers must be 6 or less. This leaves you with a very small number of linear equations (plug in $a$ $=$ 3, 4, 5, 6), which you can solve to find all the solutions. Then select the correct one.

Answer 2

Note that the square titles is of unit area !!! So the number of white tiles will be $(l-2)(b-2)$ and red tiles will be $lb-(l-2)(b-2)$ and both should be equal so by equating we will get $lb=2(l-2)(b-2)$ area must be positive.clearly l and b must be greater than 2 ! On deducing that equation we will arrive at $l+b=\frac{lb-8}{4}$, which implies {lb>8} and $lb-8=4k$ (k-constant) we get $lb=4k+8$ by substituting different values of k say 1,2,3,4 we will get lb and cross check with $l+b=\frac{lb-8}{4}$.

Answer 3

The answer is $48$. Satisfying the below condition

$$lb = 2(l-2)(b-2)$$