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ATW SATURDAY NIGHT BRAINTEASER!

By ATWadmin On December 9th, 2006

OK all you intellects out there. Aileen has once again come up with a challenge for you, this Saturday evening! Can YOU work it out?
You are off in your travels in the land of living saints and sinners, where saints only tell the truth and sinners only lies and mixers who tell the truth and lie alternately. This time you are walking to a destination, and have to pass through Talking Town. As is common with these problems, you see a fork in the road, and inquire about which way leads to Truth Town. Four people around give you advice, but you don’t know their veracity or their gender. You gather from their conversation that all four are either saints or sinners, and there is exactly one girl in the group.

The four natives make the following statements

A: Take the left fork to get to Truth Town.

B: All sinners are girls.

C: All saints are girls.

D: All people who begin their statements with "All" are either all saints or all sinners.

Just from these four statements, can you see whether the left or right fork leads to Truth Town (one and only one does)? (Also: Which one is the girl?)

Additional note the "alls" relate to the group of four and the four statements. Show your reasoning.
If I challenge your reasoning I may not be disagreeing so much as making sure that I follow you or that you have made a lucky stab in the dark!

68 Responses to “ATW SATURDAY NIGHT BRAINTEASER!”

  1. I’m pretty sure I’ve cracked this one, Aileen. And as there seem to be no takers yet, I’ll take it bit by bit…

    Can’t get anywhere starting with statement A, so leave that alone for the moment. The other three must contain some contradictions….

    On the face of it, it seems that it is possible in theory for statements (B and C) to be either both true, both false, B true and C false or vice-versa.

  2. When you say "all four are either saints or sinners", do you mean that the group, as a whole, is composed either entirely of sinners or of saints?

  3. Tom

    You see no contradiction between
    B, C

    and the givens

    "there is exactly one girl in the group.",

    "all four are either saints or sinners"?

    BTW I have a real beauty of a saints, sinners puzz;e lined up for next week or maybe the week after and another one that I havenlt worked out myself yet. Each time I leave it I have to start again.

  4. Yes, I do see it, Aileen, I just hadn’t got there yet!

  5. Ann

    each of the four is either a saint or a sinner as opposed to
    the four are all saints or all sinners.

    There are no mixers tonight.

  6. As Aileen says, if both B and C were both true, then that would mean that all the people in the group would be girls. So at least, B and C are not BOTH true,,,,

  7. Sorry Tom, you mean that until you start to analyse it, those alternatives look possible.

    I should have known that you wouldn’t have been thrown by that.

  8. What throws me the most is that none of them seem to mention shoes, shopping nor chocolate, Aileen. You’re SURE one is a girl? :))

  9. Okay, forget the question, I’ll take a stab at it anyway.

    If my reasoning’s correct C is the girl and her statement is True – All saints are girls. This would lead me to believe that the right fork leads to Truth Town.

    How I got there is another matter! And, I may well be barking up the wrong tree…

  10. Ooops, missed the answer to my question – thanks!

  11. Tom I think that is worthy of Colm’s punishment of choice ;o).

  12. Ann

    Lets have the reasoning and see if you can refrain from mentioning choclate shopping or shoes as you do?

    But what do you think of Paul Costelloe?

  13. My reasoning’s as follows

    Starting with D: if D were correct, both B and C would have to be saints as D was telling the truth.

    Therefore D is false and either B or C is telling the truth.

    If B were telling the truth, D would be a girl.

    Conclusion C is telling the truth and all others are liars/sinners.

    Am I right?

  14. Now, let’s analyse D’s statement. Moving the brackets around (am I factorising or un-factorising here? I forget!) D’s statement can be put another way:
    D: "Either B,C, and myself are all saints, or else, B,C and myself are all sinners".

    We’ve just seen that B and C cannot both be true, so the first part of D’s statement cannot be true. The only way D’s statement could still be true is if his alternative suggestion is true. (Only one of his suggestions has to be true for him to be a saint).
    But how can the second part of D’s statement be true, if he is including himself (and thus his own statement that he is a sinner) in it? A sinner cannot say "I am a sinner".

  15. So, neither of D’s suggestions is true. D is definitely a sinner. Which means that out of B and C, one is a lie and one is the truth.

    If B is the truth, then there would be at least 2 girls, as C would be the lie and D is also a liar, as above.

    So, the only alternative is that B is the lie, and C is the truth. C is the only girl, therefore the only saint. A is therefore lying; take the Right fork.
    (as Ann also concludes).

  16. I can refrain from mentioning chocolate and shoe shopping, but brain-teaser-induced migraines are another thing!

  17. Thank you that was much more eloquent!

  18. Ann

    You are making too many leaps (not saying that they arn’t accurate but you haven’t justified them

    "Starting with D: if D were correct, both B and C would have to be saints as D was telling the truth. "

    I’ll take it that this means if D is a saint then as he (gender neutral) has started with all B and C must be saints too. You need to use the fact that B and C can’t both be true. This has been established by Tom at 9:56. (I’m sure that you worked this out but I’m looking for full proof ;o) )

    "If B were telling the truth, D would be a girl."

    Conclusion C is telling the truth and all others are liars/sinners"

    Why?

  19. Tom That’s it with the proof.

    Ann you had it too. You came to the right conclusions for the right reasons. In fact it probably came so obviously to you that you didn’t realise that it needed justifying.

    As you were so quick off the mark here. I’ll give you a suplimentry….

  20. Here’s one I read the other day:

    The king is a fair and just man. You are one of three noblemen who are all vying for the hand of his daughter in marriage, and after examining their personalities and characters, the King can find no one man more worthy than the others, on balance.
    So he decides to set them all a simultaneous test of intelligence. "All three of you gentlemen, come with me to my hat-room", he tells them. They all go into the hat-room, where there are five hats on a table: three of them are white, two are black.

    All of a sudden, the king throws a blindfold over each of the men, puts a hat upon each of their heads, puts the remaining two hats out of sight, and sits the men down, all facing inwards in a tight circle, so that, once the blindfolds will be removed, each man will be able to clearly see the other two, but not himself.

    "As soon as I release your blindfolds, which I shall do all at the same time, the first one of you to tell me what colour hat he is wearing on his own head, will be allowed to marry my daughter. You may not communicate with each other in any way whatsoever. You may only call out "black!" or "white!", meaning the colour of your own hat.

    Suddenly, all the blindfolds are removed. You see two white hats before you (on the other two’s heads). A minute or two passes; no-one speaks.

    What colour is your hat, and more importantly, how did you work it out?

  21. Three friends, Alan, Bob and Cecil, sit down at a park bench. Each of them is a Saint or inner and each knows what the other two are.
    Forgetful Bob, well known for his flamboyant hats that are either red, white or blue, asks Alan: "What colour hat am I wearing today?"

    Alan replies: "There is a chance Cecil would tell you it’s blue. If you saw it, there is a chance you would say it is white."

    Bob sighs, "I should have asked Cecil – he always tells me the truth…"

    What colour hat is Bob wearing?

  22. Just seen yours Tom. I’ll have a go.

  23. Wow all hats – we must be telepathetic ;o)

  24. Ha ha! Must be to do with DV’s "Santa’s Hat Banned" post!

  25. Sorry, really slow connection tonight.

    Yes, it’s definitely easier to come up with the answer than the explanation.

    Are you by any chance a teacher?

  26. Tom

    I lkie this one. I haven;t got a clu. My excuse is that I’m not motivated t marry the King’d daughter! I take it that I don’t need to hear any guess that the other two make.

  27. A teacher me no. I wouldnt have the patience.

  28. Very seasonal – Bob’s wearing a red hat

    Alan is the saint
    Bob and Cecil are both sinners

  29. Tom

    I’m going to think out loud

    I have some notion that it has something to do with the fact that the other two know what is on your head.

    You see two white hats and that leaves one of the two black hats or the remaining white one on your head. If it is the white hat then each of the other two is also seing two white hats but you can’t know that that is what they see.

    If you are wearing one of the black hats then each of the other two are seeing a black hat and a white hat.

    Nope I am no further forward :o(

  30. Ann

    You know that you don’t get away with that on here ;o)

    Show you’re working out or it’s Colm and his punishment.

  31. Aileen, you are half way there!

  32. What was that punishment again? I recalled when it was mentioned but I’ve forgot!

    OK, if Bob is telling the truth (about C), then C also is a saint. If Bob is lying, then C also is a sinner. So either way, B and C are both either liars or sinners….

  33. I mean Saints or Sinners!

  34. Actually (back to the black or white hats), there is something rather odd about this, it’s not QUITE a pure logic problem, as you’ll see when you’ve got it!

  35. Tom

    Colm has made several comments as to people needing their bottoms smacked. It appears to be a "need" that has recently materialised or rather that he has just recognised it

  36. Who’s Colm and what’s his punishment?

  37. OOooohhh

  38. (back to Red/blue hats)…so it only leaves whether A is a saint or sinner.
    If B & C are both saints, there’s no chance their answers would contradict each other as A says, so at least one part of A’s reply is a lie, making him a liar, in that case and the hat is red….IF A is a liar.

  39. Tom

    The only other thing I can think of is that if one of the other two saw 2 black hats then he would know (or should know) that he had on a white one. but there are at most two hats on display.

    Ok if he has the white hat on the both of the other two will see both white. If one of them was to guess he would probably guess that he was wearing a black one (2 out of 3 chances),The second guy might think the same but when the first guy said black. He night think the same and reinforced by the first guess.

    donesn’t feel as if this is getting me anywhere.

  40. Errm…lets see, if B and C are both liars, Aaah, hold on, I’m stuck.

  41. Ann

    I never got to the bottom (so to speak) of it,mbut apparantly many indivduals from politicians and celebrities or other commentators are in want of a smacked bottom.

  42. Alrightie then:

    If each one knows that the other is either sinner or saint, then Alan would be able to pre-suppose that Cecil would say one thing and Bob another. This would indicate that they are both sinners. He himself hedges the question, but doesn’t lie, by saying "There’s a chance"… Bob, for his part, lies when he says Cecil always tells the truth.

  43. It’s slightly flawed, but I am trying to avoid Colm’s wrath.

  44. Ah, yes, ok, if B and C are both liars, then A’s statement cannot be completely a lie, because the hat can only be:
    Red (in which case there IS a chance that one of them will say Blue or White, in fact one or both would be bound to say at least one of those choices);
    White (in which case there is a chance that one would say blue), or
    Blue (in which case there is a chance one would say White),
    So if B & C are both liars, A must be a saint, and so again, the hat is Red!

  45. Are any of the noblemen wearing glasses?

  46. Nope, no glasses! As for me though, I am about to pour myself one!

  47. Tom

    Given that Bob and Cecil are either both saints or both sinners, what are the implications fo Alan’s statement.

    Assume that they are both saints.

  48. I did, Aileen, see 11:38

  49. Ann

    Oh go on! Live a little ;o)

    "If each one knows that the other is either sinner or saint, then Alan would be able to pre-suppose that Cecil would say one thing and Bob another. This would indicate that they are both sinners." Only if Alan is telling the truth

    He himself hedges the question, but doesn’t lie, by saying "There’s a chance"… Bob, for his part, lies when he says Cecil always tells the truth"

    Don’t forget that there is not chance that a saint would tell a lie.

  50. Okay, gotta retire. I’m an hour ahead and going boggle-eyed.

    The noblemen have got me stumped.

  51. Sorry Tom

    yes

    so if B&C are saints, then A is a sinner and the hat is red.

    If B & C are sinners
    assume that A is a saint

    Arn’t you going to pass Ann and I a glass?

  52. Nite Ann !

  53. Aileen, as I said there is a sort of quirk to the black/white hats problem. Nothing in the question is misleading or anything, but (apart from the characters, King, princes, etc) neither is anything said superfluously. One sentence in particular is crucial, but it’s not possible to say HOW crucial, on paper. Therefore the question may be impossible to phrase correctly. This would be best played as a real game, with three real contestants, if you could set it up properly.

  54. All three suitors are wearing white hats.

    Because if one of the others saw a white and black hat then he would realise that he could not be wearing a black hat as the third one would immediately realise he was wearing a white hat.

  55. Nite all – thanks for the entertainment.

  56. If no one else spoke for a minute and I shouted out "Black White " immedialtely I would be the forst to say it.

  57. Nite, Ann!

  58. Ha ha, good one, Aileen, but Ross has cracked it. The fact that "a minute or two passes; no-one speaks" is the clue.
    If your hat was black, then the other two both see one black/one white. From then, it whould only take a few seconds before one of them, realising that the other had not declared yet, cannot possibly be seeing two black hats, otherwise they’d know immediately. In which case, one of the others would be able to declare his own hat white.

  59. Ross

    I need to rewrite that maing the distintion between the hes

    "Because if one of the others saw a white and black hat then he would realise that he could not be wearing a black hat as the third one would immediately realise he was wearing a white hat. "

    If one others call him A saw a white and black hat and A had a black hat the one with a white hat, B would know that he was wearing a white hat. So he A would know that he had a white hat.

    I think!

    That’s it but how does he know that the other tow will be that bright?

  60. That’s a really good one.

    Well done Ross!

    Now back to Bob and his hat

  61. It does kinda show that the king’s logic was a bit flawed as cracking the puzzle is based on assuming that all three will be able to make the correct ogical deductions and have faith in each other to do so. He should be looking for a puzzle that will involve a differential.

  62. "That’s it but how does he know that the other tow will be that bright?"

    -Precisely, that’s the quirk about the puzzle, Aileen – time is the crucial factor, but it has to be assumed that you are slightly brighter than the others, otherwise all three would declare at once. How long would pass? A minute? 20 seconds? There’s no way of saying for sure. All you can say is that the fact that no-one declares "instantly" or thereabouts, is what proves the hats are all white.

  63. You also have to assume that they are bright enough and that they consider you and each other bright enough.

    You’re right though and the differential is getting it first.
    It’s a great puzzle.

  64. Do you play Bridge at all, Aileen?

  65. Tom

    No but like you it is something that I think I would have enjoyed. I wonder if you can learn it from a book or whether you have to learn as you play.

    I’m going to make that a New Year’s Resolution. To try an inveigle my way into some London Bridge club

  66. Come on Tom

    Bob’s hat!

    I know you can work it out.

    Or can someone else get there first?

  67. Aileen, I’ve already done it, last night.

    I started at 11:31…
    OK, if Bob is telling the truth (about C), then C also is a saint. If Bob is lying, then C also is a sinner. So either way, B and C are either both saints or both sinners.

    At 11:38 I said…
    so it only leaves whether A is a saint or sinner.
    If B & C are both saints, there’s no chance their answers would contradict each other as A says, so at least one part of A’s reply is a lie, making him a liar, in that case and so the hat is red.

    and At 11:53 I wrapped it up…
    if B and C are both sinners, then A’s statement cannot be completely a lie, because the hat can only be:
    Red (in which case there IS a chance that one of them will say Blue or White, in fact one or both would be bound to say at least one of those choices);
    White (in which case there is a chance that one would say blue), or
    Blue (in which case there is a chance one would say White),
    So if B & C are both liars, A must be a saint, and so again, the hat is Red!

  68. Tom

    Sorry

    I missed it. The thing I like abut it is that you can find out the colour of the hat but are nine the wiser about what the gentlemen in question are.

    BTW I was thinking about your hats one this morning and I had to work it through again. I must annoy them at work tomorrow with it.