Nihil,

it is not clear from your post, if you already started to write the program or just making plans. You also gave no details on your hardware - I understand that, you don't want to give clues to your opponents, but this makes it harder to help you.

So I assume you have already started with item 1 and are succesfully displaying the 81 dots.

1. don't worry, after a complete cycle of displaying switch off the beam and do everything that is needed (inputs check, algorithm, vector arrays update), then go to the beginning of the displaying sequence to start a new cycle. You can later try to make a timer-interrupt-driven display loop if you will find the picture is flickering, but for the beginning make it the easier way as described.
2.As I don't know the details, I can't help with this. Just a blind shot: Does the line display if you add it to the vector array "manually" right at the startup, not waiting for any input?
3. My approach when trying to solve an algorithmic problem is to try to visualize it and to "animate" it - e.g. doing some calculation step-by-step on a piece of paper, or draw some sketches of an evolving process. Here, you can draw the 9x9 dots playfield onto a piece of paper, get a box of matches (btw. do you know how many of them you need?), sit down with a friend and play a couple of games. Try to win, or at least try to stick to some scheme and see, if it leads mainly to wins or losses.

Good luck!

Jan Waclawek