My square patio is tiled with square tiles, all the same size. All the tiles are gray, except the tiles along the two diagonals, which are all yellow. (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between are yellow.) If there are 89 yellow tiles, how many gray tiles are there?
The solution to this problem included this language: Suppose that we have a square with e*e dimensions. The diagonals each have e tiles. If e is even, the number of yellow tiles is 2e, because the 2 diagonals don't intersect. If e is odd, then the number of yellow tiles is 2e-1. (We have to subtract 1 because the center tile is counted 2 times).
I don't understand why the number of tiles in the diagonals is 2e, or why there is no intersection between these 2 diagonals if e is even. It would be amazing if someone could explain this!
I'd love to explain.
Let's start with an even square.
On each diagonal, we're starting with a corner, then we're moving 1 row down and 1 column down.
That means the number of squares on the diagonal is the number of rows/columns.
As you can see in the above example, a 4 by 4 square has 4 yellow tiles in each diagonal.
The first tile on the (first row first column).
The second tile on the (second row second column).
The third tile on the (third row third column).
The forth tile on the (forth row forth column).
Also notice that there are 2 diagonals, with each diagonal having the same number of squares as the row/column.
So a e by e square has 2e yellow squares.
Now let's look at an odd square.
The same rule as before applies, where each diagonal has the same number as the number of rows/columns.
There are also 2 diagonals, but look at the center.
That yellow tile is shared by both diagonals, meaning we would've counted it twice in the equation 2e.
So we have to subtract 1 from it.
2e - 1
I hope this was helpful. :))