Can You Solve The World’s (Other) Hardest Logic Puzzle?

Last week , we require you to solve ‘ The Hardest Logic Puzzle In The World . ” This week , we ’re ask you to do it again – with a steel unexampled puzzle .

Sunday Puzzle #2: George Boolos’ “Hardest Logic Puzzle Ever”

UPDATE:SOLUTION

Let me begin by state the obvious : The notion that any puzzler could be the absolute , singular form , “ concentrated ” in the world is clearly ridiculous . Every trouble - solver institute unique skills to bear on any puzzle she bump , and those skills will be more well - accommodate to some problems that others . One puzzle ’s “ Hardest Puzzle Ever ” can therefore be another mystifier ’s base on balls in the park . When we prognosticate last week ’s puzzle ( the solution to which seem below)“The backbreaking Logic Puzzle In The World,”it was in reference to a version of the puzzle referred to as such by XKCD ’s Randall Munroe . Munroe ’s a sassy guy , and the puzzle is ( at least for everyone I ’ve ever spoken with about it ) , very thought-provoking . It seemed appropriate to indulge in the superlative .

https://gizmodo.com/can-you-solve-the-hardest-logic-puzzle-in-the-world-1642492269

Needless to say , I received several e - mails last week from people claiming that the title ofWorld ’s Hardest Logic Puzzle belong not to 100 Green - Eyed Dragons , but a baffling problem invented by George Boolos . Boolos was a philosopher and a mathematical logistician who taught at MIT . He was an other exponent and pioneer of provability logic , i.e. applyingmodal logicto the theory of numerical proof . He was also an self-confidence on the 19th - century German mathematician and philosopherGottlob Frege . Boolos once deliver a public public lecture explainingGödel’ssecondincompleteness theorementirely in word of one syllable . He was also an expert on puzzles of all kind , and in 1993 he reached the London Regional Final of The Times crossword competition . A smart guy , to put it mildly . In a 1996 takings ofThe Harvard Review of Philosophy , Boolos present the play along brainteaser under the statute title “ The Hardest Logic Puzzle Ever ” :

Three gods A , B , and C are anticipate , in no particular order , True , fake , and Random . True always speaks rightfully , False always speaks falsely , but whether Random speak rightfully or falsely is a completely random subject . Your task is to determine the identities of A , B , and C by asking three yes - no question ; each question must be put to precisely one god . The gods understand English , but will answer all question in their own language , in which the words foryes and no are da and ja , in some order of magnitude . You do not know which Holy Scripture means which .

Boolos also provides the follow rule of thumb :

It could be that some god gets inquire more than one head ( and hence that some god is not expect any interrogation at all ) .

What the second doubt is , and to which god it is put , may depend on the reply to the first question . ( And of grade similarly for the third question . )

Whether Random speaks rightfully or not should be thought of as calculate on the flip of a coin hidden in his brain : if the coin come down head , he speaks sincerely ; if tails , falsely .

Random will answer ‘ da ’ or ‘ ja ’ when asked any yes - no question .

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Normally , I would prefer to have solved the featured puzzle before posting it , so that I can help you , the lecturer , solve the problem without spoiling the solvent outright . However , While I have worked on it off - and - on for several mean solar day now , I have not yet solved Boolos ’ puzzle !

We ’ll be back next week with a partitioning of the solvent – and a novel puzzle ! And , as always , remember to atomic number 99 - mail me with puzzler you ’d care to see featured in future instalment !

SOLUTION to Sunday Puzzle #1: 100 Green-Eyed Dragons

The answer is that all 100 dragon plough into sparrows on the 100th midnight .

Before we take out this , get ’s consult our toolbox . dissimilar mystifier require dissimilar pecker to lick . The more puzzles one act on , the big one ’s box of tools maturate . After a while , one go seeing puzzles which , while not identical to puzzles one has solved in the past , can be worked through with strategies we ’ve swear upon in the past tense . One of the most various puppet in any puzzler ’s tool cabinet is that of restating the problem , and one of the most powerful fashion to iterate a problem is to simplify it . Many of the Sunday Puzzles featured on io9 will be made more manageable through simplification , and the character of the immature - eyed dragon is no exclusion .

So how does one simplify the 100 Green - Eyed Dragons teaser ? By making it the 1 Green - Eyed Dragon Puzzle . If you severalise a single light-green - eyed tartar that “ at least one of you ” has light-green eyes , that tartar would cognize instantly and unambiguously that she has green eye . At midnight she would change by reversal into a sparrow .

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So let ’s imagine 2 unripened - eyed dragons staring at one another , after being informed by you that at least one of them has light-green eyes . Each would look upon the other and , come across a set of green eyes , think the pursual : “ Do I have green eyes ? I do n’t do it . But if I do not , then this other flying dragon , upon seeing my non - green eyes , will fuck in a flash and unambiguously that he is the one with green oculus , and at midnight will ferment into a hedge sparrow . ” Each dragon sit and waits to see what the other does . When , at midnight , neither dragon transforms into a sparrow , each one knows instantly and unambiguously that the other dragon did not result because it , too , ascertain a dragon with green middle . And so , on the second night , each transforms into a hedge sparrow at midnight .

Through the appendage of generalisation , we reason out that any bit of green - eyed firedrake , N , will all turn into Prunella modularis on the Nth midnight following your seemingly inconsequent notice .

This can be hard to wrap your head around at first . It ’s the kind of solution that ’s liable to come across as utterly impossible until you ’ve convinced yourself of it by mould through a few more levels of installation . Even knowing the solution , it ’s prosperous to find yourself on , say , a six - tartar scenario intend “ I made this work for five firedrake , but at six it seems to come down apart ” :

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https://twitter.com/embed/status/518941015559143425

Part of this is because you ’re think about what Dragon A is thinking about what Dragon B is think about what Dragon C is call up about what Dragon D is thinking about … and so on , and it ’s easy to lose lead . We are not , after all , computers , and we are not idealized logician .

This is also why a situation like the one in the puzzle as this would almost certainly never play out in real life ; there is simply no conceivable scenario in which 100 ( non - computational ) beings could trust one another to be perfectly and infallibly logical . Daniel Leivant – a professor of computer scientific discipline and adjunct professor of math and ism at Indiana University Bloomington – illustrates the impossibility of perfect logical system AND complete confidence in others ’ logicin a introduction deliver at 1994 ’s International Workshop for Logic and Computational Complexity . In his demonstration , Leivant describes a version of 100 Green - Eyed Dragons called“The puzzler of the waterlogged children”(for all intents and purposes , an superposable mystifier to 100 Green - Eyed Dragons and Munroe ’s ‘ Blue - Eyes ’ ):

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What if my forehead is in fact clean , and the unsuccessful person of the child before me to realise that his forehead is dirty is merely due to a weak logical thinking capacity ? In that case I should not asseverate that my forehead is pestiferous . Thus my being able to make the right inference about my forehead depends on my trusting the abstract thought ability of the tiddler before me , and his count in turn on his trusting the baby before him . Also , perhaps he said “ I do n’t eff if my brow is dingy , ” not because he miss the capacity to make the good inference , but doubt the capability of the small fry before him . So the consistent approaching not only call for each youngster to be a perfect ratiocinator , but requires each kid to get into that others are too .

Again , this can be tough to enwrap your read/write head around , but it can be demonstrated mathematically , programmatically , and by pure , blunt , step - by - step , pen - and - paper induction . The logic holds fast .

Finally : What new information was added when you say “ at least one of you has green eyes ” ? For the response , we turn tothe solution leave for the Green - Eyed Dragons puzzle as I first heard it :

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Consider the compositor’s case N = 1 . Here it is clear that you provided new information , since you fundamentally severalize the one flying dragon that he has green center . But for the case N ≥ 2 , the new data is somewhat more pernicious .

Consider the case N = 2 . Prior to your announcement , A knows that B hasgreen eyes , and B knows that A has green eyes . That is the extent of the knowledge , and they ca n’t reason out anything else from it . But after you state them that at least one of them has green eyes , then A knows two things : He know that B has green eye , and he experience that B knows that there is at least one dragon with immature optic ( because A knows that B heard your information ) . B gains a similar second piece of music of entropy . This 2d small-arm of info is critical , as we saw above in the logical thinking for the N = 2 casing .

Consider the case N = 3 . A get laid that B greenish eyes , and he also have intercourse thatB cognize that there is at least one dragon with greens eye ( because A can see that B complex can see C ) . So the two bits of information in the N = 2 slip above are already recognise before you speak . What new entropy is make after you verbalise ? Only after you speak is it true that A cognise that bacillus have it away that C knows that there is at least one dragon with green eyes .

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The analogous result holds for a general number N. There is no paradox here . data is profit by your speechmaking . More selective information is added to the mankind than the information you gave . ( For example , A knows that you made your assertion while stepping onto your gravy boat and wearing a blue shirt . Or , more relevantly , A know that you made your command in front of all the other dragons . In short , it ’s not just what you say ; it ’s how you say it . ) And it turn out , as see in the cogent evidence of Claim 1 , that the new information is indeed enough to permit all the tartar to finally figure out their eye colour .

To tot up : Before you make your announcement , the following argument istrue for N dragon : A1 knows that A2 have sex that A3 knows that . . . that AN−2 bed that AN−1 know that there is at least one dragon with light-green eye . This is true because AN−1 can see AN ; and AN−2 can see that AN−1 can see AN ; and so on , until lastly A1 can see that A2 can see that . . . that AN−1 can see AN . The same result holds , of course , for any group of N − 1 dragon . The point is that it is only after you make your proclamation that the chain is extend the last tone to the Nth dragon . The fact that the Nth dragon see your statement is critical to the truth of this thoroughgoing string .

So , in the end , it become out to be of great importance how far the chain , “ Aknows that B knows that C knows that . . . ” goes . Note that if one of the dragons missed your leave-taking announcement ( which was “ At least one the 100 dragons on this island has unripe eyes ” ) , then they will all gayly remain flying lizard throughout the ages .

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In other words , your on the face of it obvious affirmation ruined everything . As IanThomasHealy put it , in my newfangled all - time favourite alternate root to this teaser :

“ Logically , the dragons adjudicate that the whole green - oculus thing is complete berth , and choose to go receive the d****-canoe who told them in the first position ” :