Concurrent Planes

This is the same as finding the number of regions $n$ great circles divide the surface of a sphere. The first great circle divides it into $2$ regions. Thereafter, the $(n+1)th$ great circle will intersect $n$ great circles in $2n$ places, creating $2n$ arcs, which creates $2n$ additional regions. Thus, the formula is

$2+2(\frac { 1 }{ 2 } n(n-1))={ n }^{ 2 }-n+2$

For $n=2014$ , it works out to $4054184$ , the last $3$ digits being $184$ .

One can check this out by tabulating the first few results

$\{ 2,4,8,14,22,32...\}$

In particular, after dividing the sphere into $8$ regions with $3$ great circles in the obvious way, note that with the $4th$ great circle can only add $6$ more regions, not $8$ .

Here's a great derivation of the formula used by Michael Mendrin. Cutting Spherical Surfaces with the Euler Characteristic