OK, I had dropped this for a while, but here are my thoughts. I haven’t scrubbed everything that could be seen through rot13 because it became excessively unreadable
For Part 1: gur enqvhf bs gur pragre fcurer vf gur qvfgnapr orgjrra bar bs gur qvnzrgre-1/2 fcurerf naq gur pragre.
Gur qvfgnapr sebz gur pragre bs gur fvqr-fcurer gb gur pragre bs gur birenyy phor vf fdeg(A)/4. Fhogenpg bss n dhnegre sbe gur enqvhf bs gur fcurer, naq jr unir gur enqvhf bs gur pragre fcurer: (fdeg(A)-1)/4. Guvf jvyy xvff gur bhgfvqr bs gur fvqr-1 ulcrephor jura gung’f rdhny gb n unys, juvpu unccraf ng avar qvzrafvbaf. Zber guna gung naq vg jvyy rkgraq bhgfvqr.
Part 2: I admit that I didn’t have the volume of high-dimensional spheres memorized, but it’s up on wikipedia, and from there it’s just a matter of graphing and seeing where the curve crosses 1, taking into account the radius formula derived above.. I haven’t done it, but will eventually.
OK, I had dropped this for a while, but here are my thoughts. I haven’t scrubbed everything that could be seen through rot13 because it became excessively unreadable
For Part 1: gur enqvhf bs gur pragre fcurer vf gur qvfgnapr orgjrra bar bs gur qvnzrgre-1/2 fcurerf naq gur pragre.
Gur qvfgnapr sebz gur pragre bs gur fvqr-fcurer gb gur pragre bs gur birenyy phor vf fdeg(A)/4. Fhogenpg bss n dhnegre sbe gur enqvhf bs gur fcurer, naq jr unir gur enqvhf bs gur pragre fcurer: (fdeg(A)-1)/4. Guvf jvyy xvff gur bhgfvqr bs gur fvqr-1 ulcrephor jura gung’f rdhny gb n unys, juvpu unccraf ng avar qvzrafvbaf. Zber guna gung naq vg jvyy rkgraq bhgfvqr.
Part 2: I admit that I didn’t have the volume of high-dimensional spheres memorized, but it’s up on wikipedia, and from there it’s just a matter of graphing and seeing where the curve crosses 1, taking into account the radius formula derived above.. I haven’t done it, but will eventually.
Part 3 looks harder and I’ll look at it later.
Part 1 is good.