Below there is a problem I invented a month or so ago. I published a link on Lesswrong and the solving began.
The first solution was 4 hexagons by LW user 9eB1. He later improved it by, quote:
right triangle that’s half the area of the equilateral triangle. You can fit 4 in the square and two in the triangle, and the score is point six
Oscar Cunningham came with this, quote:
why not get a really good score by taking a completely gigantic shape that doesn’t cover anything
This is a clever but also somehow trivial solution, we have agreed.
So he came with thi:
The thin red line is the uncovered area of the square, while the triangle can be tiled perfectly. The score is 0.57756.
Then I made I little promise, which has been broken already, that I will publish my solution on Monday, which is about two weeks ago by now.
The whole time, I was digitally evolving solutions on a computer. Just as I was not very far from Oscar’s solution he stroke again, with a much better one. More than twice as good as the previous one. This time he perfectly tiled the square and left some triangle uncovered. The score is 0.249..
This morning I almost decided to quit, when I saw something unusual on the screen. At first, I thought something wasn’t right. Then I realized what the damn computer is telling me.
The computer was evolving the following picture (visually rephrased by me for clearness).
The perfect score 0, albeit a bit trivial solution perhaps. Of course it is not necessary that you join the triangle and the square together. You can split the covering shape. Which computer did.
The problem is with the Oscar’s last solution. When I put his algorithm into the machine, it showed an error in the form of a negative result.
Well, the algorithm is okay, but 1/7 and 1/4 can’t be. But 1/7 and 1/12 can be. But the result isn’t that good.