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I think the puzzle is very clever! I just don't think the accepted answer actually fits the parameters, and I think two other answers do. The point of the puzzle is to deduce the values of the digits from the hints, and you can deduce from hint 5 (together with the promise that the puzzle is solvable) what the fourth digit is and what the other three sum to. With that information, hints 1 and 4 are redundant, and two answers are possible. The accepted answer is only unique and correct (with no redundant hints) if you ignore some of the information hint 5 gives you.

I think the clearest way I can make this point is: Whfg sebz pbafvqrevat uvag 5, jr xabj gung gur fhz bs gur svefg guerr qvtvgf zhfg or 8: nal bgure inyhr yrnirf rvgure ab fbyhgvba be zhygvcyr fbyhgvbaf. Xabjvat gung, jr pna nafjre uvagf 2 naq 3 nal jnl rkprcg "ab, ab" naq trg n fvatyr fbyhgvba rnpu jnl.

Re: "Why?", I think I can only repeat part of what I said above: Vs gur nafjre gb uvag 5 jnf "ab", jr pbhyqa'g qrgrezvar gur sbhegu qvtvg ng nyy. Gur bayl jnl jr pna or fher bs gur sbhegu qvtvg vf vs gur nafjre gb uvag 5 vf "lrf", naq gur bayl jnl guvf qrgrezvarf n fvatyr qvtvg vf vs gur fhz bs gur svefg guerr qvtvgf vf 8.

That's also why 3039 is not valid in this case: vs gur svefg qvtvg vf guerr, jr pna'g fbyir gur ceboyrz. Jr'er gbyq jr pna fbyir gur ceboyrz, fb gur svefg qvtvg zhfg or 5.

To put it another way, va lbhe bevtvany cbfg lbh rkcyvpvgyl nffhzr gung gur nafjref gb dhrfgvbaf 2 naq 3 ner obgu "lrf", ohg guvf vf whfg na nffhzcgvba, naq gur snpg gung vg yrnqf gb *n* fbyhgvba qbrfa'g zrna vg yrnqf gb gur *bayl* fbyhgvba. Vg pbhyq pbagvahr vafgrnq:

Yrg'f svk gur frpbaq naq guveq qvtvg naq nffhzr gur nafjre gb dhrfgvba 2 vf "ab" naq dhrfgvba 3 vf "lrf".

Vs gur nafjre gb uvag 5 jnf "ab", jr pbhyqa'g qrgrezvar gur sbhegu qvtvg ng nyy. Gur bayl jnl jr pna or fher bs gur sbhegu qvtvg vf vs gur nafjre gb uvag 5 vf "lrf", naq gur bayl jnl guvf qrgrezvarf n fvatyr qvtvg vf vs gur fhz bs gur svefg guerr qvtvgf vf 8.

Gur svefg qvtvg zhfg or bqq (be dhrfgvba 1 jbhyq or erqhaqnag) naq abg 1 (be dhrfgvba 4 jbhyq or erqhaqnag), fb vg'f rvgure 3, 5 be 7. 7 vf gbb ovt - gur fhz jbhyq or ng yrnfg 10 - fb bayl 3 be 5 ner cbffvoyr. Ohg vs gur svefg qvtvg vf 3, gur frpbaq zhfg or 2, juvpu pbagenqvpgf bhe "ab" nafjre sbe uvag 2. Fb gur svefg qvtvg zhfg or 5 naq gur frpbaq 0.

Fb abj jr'ir svkrq na nafjre frg (ab, ab, lrf, ab, lrf) gung yrnirf hf jvgu rknpgyl bar cva ahzore: svir mreb guerr avar.

By the same logic, the same is true of the original puzzle!

Gur rkcrpgrq nafjref (0,1,1,0,1) qb abg ryvzvangr 3139 naq 3138 nal zber guna (0,0,1,0,1) be (0,1,0,0,1) qb. 3 vf abg 1 naq 3 vf abg bqq, fb dhrfgvbaf 1 naq 4 qb abg ryzvangr vg. Jung qrgrezvarf gur havdhr fbyhgvba vf gur fnzr va nyy pnfrf: gung lbh bayl *trg* n havdhr fbyhgvba vs gur fhz bs gur svefg guerr qvtvgf vf 8. Guvf vf jung sbeprf gur svefg qvtvg gb or 3 sbe gur rkcrpgrq nafjre frg naq 5 sbe gur bguref.

> Vs rvgure bs uvagf 2 be 3 ner "ab", gur nafjref gb gur uvagf jbhyq abg havdhryl pbafgenva gur ynfg qvtvg, naq gurer jbhyq or zhygvcyr cbffvoyr fbyhgvbaf.

Naq, ntnva, gur chmmyr gryyf lbh gung jba'g unccra! Fb lbh pna nffhzr gung gur bayl jnl gur nafjref qb pbafgenva nyy gur qvtvgf vf vs gur fhz bs gur svefg guerr qvtvgf vf 8, naq gura obgu lrf/ab naq ab/lrf sbe uvagf 2 naq 3 tvir lbh rknpgyl bar fbyhgvba.

Gur rkcrpgrq fbyhgvba gnxrf nqinagntr bs guvf fnzr ybtvp: nafjref "ab, lrf, lrf, ab" gb gur svefg sbhe dhrfgvbaf yrnir cbffvovyvgvrf bs 3, 5 naq 7 sbe gur svefg qvtvg. Gur bayl ernfba lbh pna cvpx bar vf gung gur inyhr unf gb pbafgenva gur ynfg qvtvg. Ohg gur fnzr vf gehr vs lbh'ir bayl nafjrerq "lrf, lrf" gb uvagf 2 naq 3: gur svefg qvtvg pbhyq or nalguvat, ohg bayl bar inyhr jbexf.

Vs uvagf 2, 3 naq 5 ner "lrf", lbh pna nyernql qrqhpr nyy sbhe qvtvgf! Lbh'ir tbg k23l, lbh xabj l > k+2+3, naq *gur chmmyr gryyf lbh gurer vf bayl bar fbyhgvba*. Fb k+2+3 pnaabg or yrff guna 8, be gurer jvyy or zhygvcyr cbffvovyvgvrf sbe l, naq vg pnaabg or zber guna 8, be gurer jvyy or ab cbffvovyvgvrf sbe l.

V pnzr onpx gb rqvg zl zrffntr jvgu fbzrguvat gb gur fnzr rssrpg, zl ncbybtvrf :) Vg bppheerq gb zr gb gnxr zl bja fhttrfgvba frevbhfyl. Vs 2, 3 naq 5 jrer gur bayl uvagf, gura gur "lbh unir rabhtu vasbezngvba" cebivfb jbhyq sbepr gurz nyy gb or "lrf" naq yrnq gb n fvatyr fbyhgvba. Va gung frafr dhrfgvbaf 1 naq 4 ner vaqrrq erqhaqnag. Juvpu znxrf zr ernyvmr: lbhe cbvag zrnaf gung gur npprcgrq nafjre vf gur bayl bar bs gur guerr gung'f _abg_ pbeerpg!

I love this puzzle, but I think it's not quite correct: three solutions are compatible with the specifications. Spoilers and discussion in rot13:

Gur svefg naq sbhegu dhrfgvbaf zhfg or nafjrerq "ab", gb nibvq erqhaqnapl, naq gur svsgu zhfg or nafjrerq "lrf" gb trg rabhtu vasbezngvba nobhg gur sbhegu qvtvg, fb jr pna pbafvqre whfg uvagf 2 naq 3.

Gur npprcgrq fbyhgvba pbzrf sebz nafjrevat obgu "lrf". Nafjrevat obgu "ab" qbrfa'g tvir rabhtu vasbezngvba - vg yrnirf, rt, 3419 naq 3149 nf cbffvovyvgvrf. Ohg jr pna nafjre 2 jvgu "lrf" naq 3 jvgu "ab" naq trg n fbyhgvba bs 5219, be nafjre 2 jvgu "ab" naq 3 jvgu "lrf" gb trg n fbyhgvba bs 5039.

Obgu pnfrf yrnir ab erqhaqnag dhrfgvbaf, naq va obgu pnfrf xabjvat gur svir nafjref vf rabhtu gb svaq bar fbyhgvba.

Gurer'f n ceboyrz jvgu guvf ernfbavat, naq gb zr vg oernxf gur chmmyr: gurer'f ab ernfba lbh unir gb nafjre dhrfgvbaf 2 naq 3 jvgu "lrf". Va snpg vs lbh nafjre gurz "ab", "lrf" naq "lrf", "ab" erfcrpgviryl (jvgu gur bgure nafjref erznvavat gur fnzr) lbh trg gjb qvssrerag nafjref: 5039 naq 5219. Obgu fngvfsl gur pbaqvgvbaf bs gur chmmyr: xabjvat gur nafjref gb gur dhrfgvbaf tvirf lbh gur fbyhgvba, naq erzbivat nal dhrfgvba yrnirf lbh jvgu zhygvcyr fbyhgvbaf.

Dhrfgvbaf 1 naq 4 nera'g erqhaqnag: gel fbyivat gur chmmyr jvgubhg gurz!