In the diagram below, each row (reading from left to right) and column (reading from top to bottom) gives instructions for painting one of the 12 pentominoes:
Sometimes an idea languishes in one of my notebooks for a few years before I can come up with the right iteration of it. My original idea here was to use a 4×4 grid. That would give me 8 pentominoes, (perhaps 10 using diagonals) but elegance surely requires all 12 to be present.
A combination of circumstances led me back to this problem. Some friends of mine have a tradition of playing RoboRally on New Year’s Day every year. This is a board game where you use cards with arrows on them to instruct your robot to move around a grid of squares. Also, in returning to the magic 45-omino problem, I was considering grids that could be used in sparse magic squares.
It might be possible to make an interesting grid puzzle, along the lines of sudoku, using this kind of grid as a basis. Most of the grid would start empty, except for a few squares in which arrows would be given at the start. Then the solver would fill in the rest of the grid by logical deduction so that the horizontal and vertical lines contain instructions for paining all of the pentominoes as above. Since the grid would have significantly fewer squares than a sudoku, this puzzle might be quicker to solve, but that doesn’t mean that it would necessarily be less interesting.