Packing circles into circle of diameter 7 How many unit circles can you fit inside a circle of diameter 7 such that no circle overlaps any other circle? Please explain the concept or any tricky process regarding this problem.
 A: As David already pointed out in a comment, Erich's Packing Center is a very useful resource for this kind of question. Regarding circles packed inside a circle, you'll find that the smallest circle to contain $8$ unit circles has radius $3.304{+}=1+\csc\frac\pi7$ (see also A121598) whereas the smallest to contain $9$ has $3.613{+}=1+\csc\frac\pi8$ (see also A121601). Your $\frac72=3.5$ is between these two, therefore $8$ would be the maximal number you could fit in.
Notice that EPC annotates results as “Trivial”, “Found by … in …” and “Proved by … in …”. Both the results above fall in that last category, so they are not only the best packing known today, but in fact proven optimal. The authors mentioned are “Braaksma in 1963” and “Pirl in 1969”. I haven't found a publication matching that first reference, but Pirl does cover that case as well, in his German text.
Regarding the original title of your question (which is pretty unspecific so perhaps it should be edited):

explain the concept or any tricky process regarding this problem

Showing that you can fit $8$ circles is easy. The core idea of showing that you can't fit $9$ lies in considering the smallest circle where you can fit $9$. Pirl argued that you can't get an optimal arrangement where less than $8$ circles are arrayed on the outside of the figure. So $8$ circles outside and one inside is optimal, and the radius if all the outside circles touch is the one quoted above, which is larger than your circle.
