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.
I'm not sure I know what you're getting at because:
>But if the first digit is 3, the second must be 2
Why?
>So now we've fixed an answer set (no, no, yes, no, yes) that leaves us with exactly one pin number: five zero three nine.
This is not correct, because with (no, no, yes, no, yes) the pin 3039 would also be valid, so five zero three nine is not a unique solution.
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.
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.
>Just from considering hint 5, we know that the sum of the first three digits must be 8: any other value leaves either no solution or multiple solutions. Knowing that, we can answer hints 2 and 3 any way except "no, no" and get a single solution each way.
I think this is where both our reasonings are "the other way round": With this being a Constraint Satisfaction Problem without an a priori fixed set of constraints, you're instead treating this more like an iterative deduction based problem?
Also sorry if you've already stated that elsewhere and I didn't catch that, but what exactly is your actual problem with this puzzle? I mean it clearly has a single unique solution (ie. exactly 1 set of answers that results in exactly 1 pin) which you can proof by just bruteforcing all permutations of pin numbers and answers.
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.
Just to see if I'm clear on the disagreement here: it's about whether or not it's valid to use "there is a unique solution" as part of the constraints?
Because I had started by assuming that since hint 5 is the only thing that constrains digit four, in order to have a unique solution, the first three digits must sum to either eight or zero. And that still seems obviously true to me.
But your position is that "there is a unique solution" is sort of an external check and may not be used as an actual constraint? I can see that if you're not allowed to say "there is a unique solution, therefore the digits must sum to 8," then you're left with "IF the OTHER hints fail to force the digits to sum to 8, THEN you don't have a unique solution." And that makes hints 1 and 4 are NOT redundant in the yes/yes/no case for hints 2/3/5.
Do I have that right?
It still does seem very strange to me that unique solution would not be an allowable constraint when non-redundant hints is. So yeah, seems like a "both reasonings are the other way around" problem, and therefore a poorly-designed puzzle. But hey, game jam.
Ah, dustbunnies. Now I see the error in my logic. It's true that hint 5 says that ifthere is a unique solution, it must have this property. But that doesn't help you answer the question of whether a particular set of hint answers rules out all of the non-unique answers that may or may not have that property. Gah. OK.