MHB Solving a problem using a simplified version

[estex198]

While I know the answer to this problem, I can't figure out *how* to get it.

Carmelo has been commissioned to create a decorative wall for the 21st Annual X Games consisting of a square array of square tiles in a pattern forming a large X. The following example shows a pattern with 5 rows and 5 columns. If the wall will have a similar pattern with 21 rows and 21 columns, how many of the colored tiles will be needed?

View attachment 4368
​

With the 5x5 array there are 25 total blocks and 15 colored blocks. I thought maybe I could use ratios and cross multiply: 15/25 = x/441, but this isn't giving me 79. I know 79 is the correct answer not only because its shown in the back of the book, but because I actually took a ruler and completed the pattern on paper (tedious as it was, I was really frustrated and needed to know I wasnt wasting my time.) Please help!

 

[MarkFL]

We know we will have the top and bottom rows colored for a total of 42 tiles. Then one diagonal will be 19 (21 minus the two already counted) and the other diagonal will be 18 for a total of 42 + 19 + 18 = 79. :D

Using this same logic, we could derive a formula for an $n\times n$ grid (where $3\le n$ and $n$ is odd). If we let $C(n)$ denote the number of colored blocks for such a grid, we would get:

$$C(n)=n+n+(n-2)+(n-3)=4n-5$$

 

[estex198]

Rusty: So your logic is that since two rows will be filled in (n + n), the diagonals will equal the remaining number of lines minus two, with the exception that one of the lines will be one less (n-2) + (n-3), since it will intersect with the other line. Thanks a million btw for a clear and direct answer!
I'm still curious however, if there is a more formal method of solving this problem? Can anyone explain why cross multiplying ratios will not work? Thanks in advance!