Using the image above, the rules are every dot must have a line through it, no diagonal lines, no crossing over lines, no backtracking, no going out of bounds, and no picking up the pencil to start at another location. A friend says it is solvable and I want to say it isn't. Is this puzzle solvable under these set of rules? If yes, post your solution.
No, it isn't solvable.
The proof is as follows.
Colour the dots black and white in a checkerboard pattern. Then you have 8 dots of one colour and 10 of the other, say 8 white and 10 black (the upper right and bottom left dots being black). But at each step you must pass either from a black dot to a white dot or vice verse, so any valid path must pass through the same number of dots of both colours, give or take one. So there's no valid path passing through 8 and 10 dots of the two respective colours.