But maybe there are even more optimizations.ĭoh! I screwed up the calculations for that improved method, and it gives 1.000039506. But it's possible I screwed up somewhere, and it looks like there are other smaller optimizations that I could make.Īha! I have a better method. I worked out some equations then solved numerically, and my maximum size seems to be 1.000039015 cm. The only reason I know I didn't screw up the equations is that I didn't use any. That sounds like it's in the right ballpark. On the other hand I could have easily screwed up the equations. I get a possible diameter of 1.000039505 for an 8*4 solution. I think I might also be able to get James' solution, but the numbers are a bit tricky. Perhaps we have different solutions for the configuration of the circles that just happen to have the same M*N I can see that my solution can be trimmed to 7*8, though surprised by the slight difference in diameter wrt towr's solution.
« Last Edit: Jul 3 rd, 2003, 7:22am by towr » I'm still quite a way off from James' 8*4 solution, but I have a 8*7 solution as well now. « Last Edit: Jul 3 rd, 2003, 6:17am by James Fingas » Here's a related question: What if M,N were real numbers? I've experimented with other strategies, but I think this one is optimal.Īs for an exact diameter, I don't have one, but I know you can make them at least 1.00000235 cm in diameter (quite conservative-a better calculation is too messy for me right now, but probably you could get 1.00001 cm diameter). The M*N circles can be larger by a very very small amount. I did a calculation that was evidently wrong. The distance (d) between circles in the same row is about 0.05733373693Įach rows adds about 0.8559664362 to the height (h),Īnd this time there's also enough extra room left on the side to actually fit the even rows.
MxN=9x7 I get a maximum diameter (2r) of 1.008234938Īnd it fits 8 rows of 8 circles (again one more than needed) I think there was still an error in my last result, « Last Edit: Jul 3 rd, 2003, 2:31am by towr » The diameter wasn't asked, but if you want it this should give enough information to find it, somehow.Īnd yes, incidentally it also fits one circle more. It's easy to calculate that there is actually space left at the top, and since 7 is smaller than 8 there is space in the length-direction as well. And there is also room left at the top, so that the circles can be bigger since they can expand in height, and length. If you take a length 8, height 6 box, you can put 7 layers of 7 (diameter 1) circles in it. And in my case I think it does, and I'm sure that's also the case in James' solution. Just because I don't know the diameter doesn't mean larger circles don't fit. « Last Edit: Jul 2 nd, 2003, 4:37pm by Leo Broukhis » What is the smallest value of M*N such that an M cm by N cm rectangle can contain more than M*N identical circles, each with diameter 1 cm. ĭoc, I'm addicted to advice! What should I do?Īs neither towr, nor James specified the diameter, I tend to believe that they were solving a different problem: Wikipedia, Google, Mathworld, Integer sequence DB « Last Edit: Jul 3 rd, 2003, 3:33am by towr » I think 8*6=48, but I'm not quite sure, since my numbers change every 10 minutes (had lots of errors in my method). Some people are average, some are just mean. « Last Edit: Jul 2 nd, 2003, 9:03am by Leo Broukhis » All circles must fit entirely within the rectangle at the same time, no circle may overlap any other circle.Ĩ4? And the diameter is 1.01036297108184508789. What is the smallest value of M*N such that an M cm by N cm rectangle can contain M*N identical circles, each with diameter greater than 1 cm? If M and N are integers, it is easy to fit M*N circles of 1 cm diameter into an M cm by N cm rectangle without the circles overlapping. Topic: Fitting Circles In A Rectangle (Read 16254 times) Hard (Moderators: Grimbal, Eigenray, william wu, Icarus, ThudnBlunder, SMQ, towr) RIDDLES SITE WRITE MATH! Home Help Search Members Login Register « wu :: forums - Fitting Circles In A Rectangle » Maths week treasure hunt.Wu :: forums - Fitting Circles In A Rectangle
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