Question

I have tiled my square bathroom well with congruent square tiles. All the tiles are red, except those along the two diagonals, which are blue. If I used $121$ blue tiles, then the number of red tiles, I used is:
A. $900$
B. $1800$
C. $3600$
D. $7200$

Answer 1

Hint:
Let the dimension of the square bathroom be $n \times n$ and now we have two cases that if $n$ is even then the number of diagonal tiles$ = 2n$
But if $n$ is odd then the number of diagonal tiles will be $ = 2n - 1$ so now we can find the number of red tiles.

Complete step by step solution:
Here in the question we are given that we have tiled the square bathroom well with congruent square tiles. All the tiles are red, except those along the two diagonals, which are blue. If I used $121$ blue tiles, then we need to find the number of red tiles used.
Here congruent means that tiles are of the same area, same perimeter, same length and the thickness.
In the diagonal we have used the blue tiles and the number of blues tiles is given in the question which are $121$ then we need to find the number of red tiles that are present in the bathroom.
Now let us suppose that the dimension of the square tile be$n \times n$ that means $n$ square plate in the length and $n$ square plates in the breadth.
In case 1
We have an odd number of square tiles. So if we have $n$ as odd then the number of diagonal square pates is given as $2n - 1$ and we are given that in the diagonal there are $121$ blue tiles so we cans say that
$
\Rightarrow 2n - 1 = 121 \\
\Rightarrow 2n = 122 \\
\Rightarrow n = 61
 $
Therefore total number of tiles used$ = {n^2} = {61^2} = 3721$
Number of red tiles will be $ = {\text{total number of tiles used}} - {\text{number of blue tiles used}}$
$ = 3721 - 121 = 3600$
For case 2
If we have $n$ as even that is even number of tiles then the number of diagonal tiles will be $2n$
So $
\Rightarrow 2n = 121 \\
 \Rightarrow n = 60.5
 $ but we know that n cannot be in fraction or decimals as it denotes the number of tiles which must be an exact number so we can say that $n$ cannot be even.

Therefore the number of red tiles $ = 3600$.

Note:
If the dimensions are assumed to be $n \times n$ and it is odd then the number of diagonals must be odd but if the dimensions we assume are even then the number of the diagonal square plate must be even.