Remember the Dots and Boxes puzzle? Of course you do.
Before I give away the whole solution, I will state the targets. For a 5x5 grid, the target is 15 lines. For a 6x6 grid, the target is 23 lines. For a 7x7 grid, the target is 31 lines. If you show me a solution that is better than any of these targets, I will be amazed, because I just don't think it's possible to do any better.
Solution to the 5x5 grid
You should know, by the way, that the reason I reveal these one at a time is in case anyone is inspired to go back and try to solve the larger grids. Eduard Baumann was the first solver of the 5x5 grid, and Secret Squïrrel was the first solver of the 6x6 grid. Nobody reached the target for the 7x7, but you still have a chance to try it yourself.
Solution to the 6x6 grid
There is another distinct solution to the 6x6 grid using the same number of lines.
Solution to the 7x7 grid