Problem Solving

Q. A rectangular grid of 182 squares. Each row in the grid has 14 unit squares and each column in the grid has 13 unit squares. The number of unit squares through which a diagonal of the grid passes is :-
A. 26
B. 20
C. 14
D. 30

Funny question. My first instinct was to use the Pythagorean theorem, but wait the diagonal of the rectangle goes through squares diagonally.

If the rectangle were a 14 x 14 square, the diagonal would go through 14 squares, but it's not. So the diagonal doesn't quite hit the corner of the squares.

I guess I'll hack it and say that as the diagonal goes through each square, it does not quite make it all the way to the second corner, going through one side of the square at a point 13/14 sides length from where it started.

So for the first square, a diagonal going left upper to right lower of the rectangle through the 14 columns starts at the top left corner of the square and hits the right side 13/14 of the way to the bottom corner.

Went through 1 square.

For the second column the diagonal hits at 1/14 up from the upper corner of the second row square and 2/14 up from the lower corner.

Went through 2 squares.

Third column, 2/14 above upper and 3/14 above lower.

Went through 2 squares.

Fourth, 3/14 and 4/14.

Went through 2 squares.

OK. I see the pattern. It goes through two squares for each of the 14 columns except for the leftmost and rightmost, in each of which it goes through 1 square.

1 + (2 x 12) + 1 = 26

The correct answer is A.