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.

EDIT:

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.