# Thomas comments on Open thread, March 13 - March 19, 2017

• In­ter­est­ing. We will see where this is go­ing to go.

• Any luck? I’d be in­ter­ested in see­ing some of the com­puter solu­tions even if their scores didn’t beat mine.

By the way I can now im­prove my score to 14sqrt(3)-24 = 0.249… . My cov­er­ing shape is a 14 by 17 right-an­gled tri­an­gle. This clearly tiles the square perfectly and you can also fit 24 of them into the equilat­eral tri­an­gle. To see this first di­vide the equilat­eral tri­an­gle ex­actly into 24 right-an­gled tri­an­gles of sides 14 and 1/​(4sqrt(3)), and then note that 17 < 1/​(4sqrt(3)). There’s no point in draw­ing a pic­ture since you can barely see the gaps.

• The cheaty solu­tion at the end de­pends on what seems to me an un­in­tended in­ter­pre­ta­tion of the ques­tion (though, given that the same per­son wrote the ques­tion and the pro­gram that found the solu­tion, maybe my idea of what’s in­tended is wrong). I took “tile both poly­gons” to mean “tile poly­gon 1 AND tile poly­gon 2″, not “tile the union of poly­gons 1 and 2”.

• It is a solu­tion similar to the one with the big shape which doesn’t cover any­thing, but the re­main­der is ar­bi­trar­ily minis­cule rel­a­tive to the shape.

We called it “a triv­ial solu­tion”, as I call this solu­tion triv­ial, but maybe less triv­ial, since it ac­tu­ally cov­ers all and it wasn’t ex­plic­itly for­bid­den, not to peo­ple, not to the com­puter.

Now, I told my chitin’ comp, don’t do that, each in­stance of the shape should cover ei­ther the tri­an­gle, ei­ther the square!

We will see.

• I must say, that this solu­tions of yours is quite im­pres­sive. Quite im­pres­sive in­deed.

• I promise you scores and images of solu­tions, what­ever they will be. Calcu­la­tions are un­der way right now and they should be available soon.