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22 votes
1 answer
1k views

Does greedy circle packing exhaust the measure of every bounded open set in the plane?

The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered. The greedy circle packing of a ...
Joel David Hamkins's user avatar
7 votes
1 answer
225 views

Optimal stacking of split logs

Consider firewood logs as unit-radius cylinders of the same length. Each log is split into $k$ pieces by equiangular sectors meeting in the circle center: $k=2$ leads to semicircles, $180^\circ$; $k=3$...
Joseph O'Rourke's user avatar
2 votes
1 answer
230 views

Some inequalities on chain of circle packing

By my computation, I pose a conjecture as follows and I am looking for a proof: Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...
Đào Thanh Oai's user avatar
2 votes
1 answer
127 views

On covering a disk by non-overlapping subdisks

I posted this question many years ago on math stackexchange but it did not get an answer. It had circulated as a puzzle in graduate school. A disk $D$ of radius $1$ contains disks $D_i$ ($i \ge 1$) of ...
coffeemath's user avatar
1 vote
1 answer
322 views

Settling a circular argument: room for one more?

By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller ...
Gmackematix's user avatar
1 vote
0 answers
202 views

Some Problems On Apollonian Gasket

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems: Let three $(A)$, $(B)$, $(...
Đào Thanh Oai's user avatar