Answer to old chestnut: connect 3x3 points with 4 lines:

There is a slight trick to the solution. You have to allow lines to go beyond the edge of the square containing all the points in order to make some of the lines connect, and some people just don't think of this as a possibility. But here's how it's done:

       /|
      / |
  .  .  .
   \/   |
   /\   |
  .  .  .
 /    \ |
/      \|
--.--.  .

For larger grids, you can extend this solution with a spiral. Begin by extending the line through the main diagonal in the solution above by one more unit to the upper left (so that it reaches the next grid point -- but this is not one of the points we will use in the 4x4 solution!). Then add two lines, one straight down for four units, then one to the right, and you'll have a solution for a 4x4 grid.

|\      /|
| \    / |
.  .  .  .
|   \/   |
|   /\   |
.  .  .  .
| /    \ |
|/      \|
.--.--.  .
|
|
.--.--.--.

By continuing to wrap this line around the figure you can get solutions for any size square grid.

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