In the 1970s, the German fountain pen manufacturer Pelikan ventured into the budding market of authored board games with a series of games in a well designed format. The market wasn’t ready, and some of the games had serious issues with their rules. I bought three of them, and only one was a keeper: Diamant, by Andrea Steyn.
You got a game board (advertised as 40mm thick…) to hold 12 silvery plastic mirrors and 16 small diamonds in 4 colors, as well as a set of goal cards. Players would take turns placing mirrors on the board, or later moving them. Then they could send (virtual) light beams from their side of the board, having them reflect on the mirrors to eventually hit one of the diamonds, which the player could then take in order to work towards completion of their personal goal card.
Above is a top view of the board, showing mirrors and color coded light beams. The numbers at the border indicate how many mirrors are hit by the light beam, starting/ending at that number.
By discarding the colors and the mirrors, you can turn any mirror configuration into a puzzle, like the one above. A good strategy is to place mirrors that give the correct small numbers. While not forced, a first step towards a solution might look like this, dealing with the 2s:
Now you can see which of the 3s have to connect, and from there the solution is easy to find. Below is a much harder puzzle on a 6×6 board. My solution uses 15 mirrors of each kind.
I don’t know how (computationally) hard these problems are, but they are obviously NP (non-deterministically polynomial): It is trivial to check whether a solution is correct. There are many variations of this simple puzzle (building in refraction, for instance…), and cute little math problems one can ask about the numbers that can show up at the border. Maybe I’ll come back to this in the future.
The Inner Frame 