No one sent us a complete solution to this problem, but here is the start of a solution which shows some good mathematical thinking and a systematic way of working.

We decided to investigate what happens if you keep the $x$ coordinate the same and change the $y$ coordinate. Here is our table to show the number of squares and gridlines crossed when the $x$ coordinate is $5$.

We noticed a pattern for the number of squares crossed, which works for every $y$ coordinate we tried except $5$. If you add together the $x$ and $y$ coordinates and take away 1, you get the number of squares crossed. This is because for example to get from (0,0) to (5,3) you go along 5 squares and up 3 squares, meaning that you travel through 8 squares altogether, but that counts the corner square twice so you need to take away 1.

It doesn't work for (5,5) because you go diagonally through the corners of the squares instead of cutting through the edges.

Can anybody build on this thinking and explain the patterns found?