obgu

Fvapr F_3 pbagnvaf obgu N naq O, G_1 naq G_2 obgu pbagnva obgu N naq O.

Well I think you mean: (rot13) Yvtug ebcr 1 ng bar raq, naq ebcr 2 ng obgu raqf.

Rnpu zbivr vf pbaarpgrq gb gur bar rvgure fvqr bs vg ol na npgbe jub fgneerq va obgu.

Vg qvivqrf gur fcurer vagb gjb urzvfcurerf, jvgu gur gjb cbvagf orybatvat gb obgu bs gurz.

Abj qrsvar G_1 = F_1 H F_3 naq G_2 = F_2 H F_3; ol vafcrpgvba, gurfr ner obgu pybfrq urzvfcurerf.

Ner obgu urnegf ba rvgure fvqr bs gur oynax cbvagvat va gur fnzr qverpgvba (abegu fbhgu rnfg jrfg hc qbja)?

It's interesting to note that at the core of Asimov's Foundation (spoiler: Va n frafr, ng gur pber bs obgu bs gurz.

Ubjrire, gur rssrpg bs yrneavat gur nyvra ynathntr vf pbzcyrgryl qvssrerag va obgu fgbevrf.

Grra Erl frrzf gb unir orra nonaqbarq naq gb yvir ba ure bja, ohg fur vf nyernql irel pbzcrgrag va obgu grpuabybtl naq svtugvat.

Ybbxrq ng gur ovbtencul bs obgu crefbaf naq Ubyorvaf bgure cvpgherf.

Ol flzzrgel, jr fubhyq rkcrpg gur zbivrf jvgu uvturfg pevgvp fpberf gb unir ybjre nhqvrapr fpberf, naq ivpr irefn va obgu pnfrf sbe ybj/uvtu nhqvrapr fpber.

Answer (ROT13): Vg qrcraqf ba gur cnevgl bs gur ahzore bs fdhnerf, juvpu qrcraqf ba jurgure gur qvzrafvbaf bs gur tevq fvqrf ner obgu bqq be abg.

Frg sebz pbagnvaref; gurl obgu fhccyl n "frg" nofgenpg qngn glcr, ohg qvssre va gur qngn fgehpgher hfrq gb vzcyrzrag vg (unfuzncf if.

Sbe rkrzcyr: 3,X Pyhof, 8 Fcnqrf, 10 Qvnzbaqf, X Urnegf Fb V erzbir "3 Pyhof" naq qvssrerapr vf 3 (fb V beqre gur erznvavat pneqf va gur guveq beqrevat beqre 213) ohg +3 naq -3 obgu tvir n ybjre inyhr guna X (3 be 10).

Obgu vaibyir gur neeviny bs nyvraf ba rnegu naq qrgnvy n yvathvfg'f nggrzcg ng haqrefgnaq gurve ynathntr naq ubj yrneavat gung ynathntr nssrpgf gung yvathvfg.