Well I have been very busy at work for the last few months and have only just managed to find a little spare time to look at the Python Challenge.
However in the last day I have managed to find a few moments to play with Level 31 of the challenge and with the help of a lot of hints from Nic, from the Python Challenge Forums, I have finally managed to complete this level. Here is the ROT13'ed solution:

Guvf yriry jnf n guerr fgntr nssnve, jvgu obgu gur svefg naq ynfg fgntrf orvat 'Tbbtyr' onfrq.

Fgntr 1

Jr ner fubja n cvpgher bs n engure cunyyvp ybbxvat ebpx sbezngvba. Vs jr pyvpx ba gur vzntr, jr ner nfxrq sbe n Hfre/Cnffjbeq pbzovangvba. Gur urnqvat sbe gur cntr nfxf 'Jurer nz V?'. N yvggyr tbbtyvat sbe Tenaqcn naq Ebpx fubhyq yrnq lbh gb vasbezngvba ertneqvat guvf ebpx naq gung vg vf ybpngrq va Gunvynaq. Or pnershy gubhtu, nf V jnf pnhtug bhg ba gur rknpg fcryyvat naq fgehpgher bs gur Hfre cneg. Gur erdhverq qrgnvyf ner:

Hfre = xbufnzhv
Cnffjbeq = gunvynaq

Fgntr 2

Guvf yrnqf hf ba gb yriry gjb bs gur chmmyr, jurer jr ner fubja n TVS bs n Znaqryoebg frg naq ner nyfb cebivqrq jvgu fbzr nqqvgvbany vasbezngvba jvguva gur UGZY cntr fbhepr, znvayl:

jvaqbj yrsg="0.34" gbc="0.57" jvqgu="0.036" urvtug="0.027" bcgvba vgrengvbaf="128"

Gurfr inyhrf ner nyy irel vzcbegnag gb cresbezvat n Znaqryoebg pnyphyngvba, gur vgrengvbaf inyhr ersref gb gur znkvzhz ahzore bs vgrengvbaf gb gel naq gur bgure inyhrf qvpgngr gur jvaqbj gb cresbez gur pnyphyngvbaf jvguva v.r. Gbc Yrsg = 0.34, 0.57 Obggbz Evtug = 0.34 + 0.036, 0.57 + 0.027. Lbh nyfb unir gb ybbx pybfryl ng gur znaqryoebg.tvs, nf jr arrq gb pbzcner bhe erfhygf ntnvafg gung vzntr.

Abj V qb abg qb guvf irel bsgra (npghnyyl abg fher V rire unir, ohg V nz tbvat gb cebivqr zl pbqr sbe guvf engure guna rkcynvavat ubj gur znaqryoebg frg jbexf, guvf vf cnegyl orpnhfr lbh pna svaq gung vasb sebz Jvxvcrqvn.bet ohg nyfb orpnhfr gurer ner n ahzore bs 'nqwhfgfzragf' arrqrq gb trg guvf gb jbex:

vzcbeg Vzntr

vzt = Vzntr.bcra('znaqryoebg.tvs')
vzt = vzt.genafcbfr(Vzntr.SYVC_GBC_OBGGBZ)

k_fgneg = 0.34
l_fgneg = 0.57
qvssf = 0

sbe l va enatr(0, 480, 1):
sbe k va enatr(0, 640, 1):

n = k0 = k_fgneg + 0.036 * k / 640
o = l0 = l_fgneg + 0.027 * l / 480
ybbc = 0
vg = 0

juvyr ybbc == 0:
vg += 1
p = n
n = n * n - o * o + k0
o = 2 * p * o + l0

vs vg == 128:
ybbc = 1

vs n * n + o * o >= 4:
ybbc = 1

tvs_cvk = vzt.trgcvkry((k,l))

vs tvs_cvk <> vg naq tvs_cvk < 127:
qvssf += 1

cevag qvssf

Abj ehaavat qbja gur pbqr, gur znva cbvagf bs vagrerfg gung ner nobir naq orlbaq gur Znaqryoebg frg ner nf sbyybjf:

vzt = vzt.genafcbfr(Vzntr.SYVC_GBC_OBGGBZ)

Guvf vf erdhverq orpnhfr gur vzntr jr ner pbzcnevat jvgu vf n ubevmbagny zveebe vzntr bs gur Znaqryoebg lbh perngr hfvat gur inyhrf cebivqrq.

vs tvs_cvk <> vg naq tvs_cvk < 127:

Gur tvs_cvk < 127 jnf erdhverq orpnhfr gurer jrer ab 128'f va gur TVS, gurl jrer nyy 127 ohg gurfr jrer abg npghnyyl n qvssrerapr jr jrer ybbxvat sbe.

Nsgre ehaavat guvf pbqr, jr svaq gung gurer ner 1679 qvssreraprf orgjrra gur pnyphyngrq Znaqryoebg frg naq gur TVS vzntr. Guvf yrnqf hf vagb Yriry 3.

Fgntr 3

Onpx gb tbbtyr ntnva. Abj gur svany vgrz bs vasbezngvba jr erdhver vf nyfb va gur UGZY fbhepr naq gung vf 'HSBf?'.
Chggvat 1679 naq HSBf vagb tbbtyr fubhyq oevat hc fbzr vagrerfgvat vgrzf bs vasbezngvba, vapyhqvat fbzr nobhg n genafzvffvba sebz FRGV bs n fgevat bs 1679 ovanel chyfrf. Jurer jrer gurfr chyfrf frag gb? Nccneragyl n zrffntr pnyyrq gur 'Nerpvob zrffntr' jnf frag gb tybohyne fgne pyhfgre Z13 va 1974 naq 'nerpvob' vf gur nafjre gb guvf Chmmyr naq gur ybpngvba bs gur arkg yriry.