I think it’s clear that it can’t be cat. Bernard says "Now I know the secret word. As in, I didn’t before, but now I do. Therefore Cheryl knows that her “a” could not have been from a “has”, because then Bernard would have gotten “h” or “s”, and would not have had to wait for Albert to answer before figuring out the answer. Which would mean that she didn’t have to wait for Bernard, and also wouldn’t say “*Now *I know the secret word.”
Oh good, words! I like this one. Thanks!
Apologies!!! The reason you are having trouble is that I screwed up badly, and the puzzle as I first stated it has no solution! If you change it so that the two different numbers are known to be strictly between 1 and 100 (rather than 1 and 50), then there is a unique solution as intended (even though both integers in the solution are less than 50).
In order for A to make her initial declaration, the sum has to be such that, no matter how you split it into two addends, the respective product cannot be factored in a unique way. So two distinct primes are out (for example, 5=2+3 is bad), but so are splits like 6=2+4 which give the cube of a prime or the fourth power of a prime.
In my erroneous formulation, 41 would be out, because eg 41=10+31 yields the product 310=5x62=10x31, but 62 is greater than 50, so you see the subtlety. 2x39 = 3x26 would be fine anyway.
82 is out because 82=35+47, even though not both addends are primes, and similarly 51=17+34 is out since 17x34=2x289 but 289 is too big.
Actually going through all possible products for each sum and crossing off the bad ones, I am left with only 11, 17, 23, 27, 29, 35, 37, 41, 47, and 53 as sums that enable A to deduce that B cannot know the numbers. (In the erroneous version, the list ends at 29.)
[spoiler]But he doesn’t say the “I didn’t before” part; just the “now I do” part.
Again, that’s not true of other stuff in this thread; the OP, for example, has someone painstakingly and explicitly spell out that “I didn’t know the answer before, but when you said that, I realized that, in fact, I do know the answer." And, in a later post, we’re given an example where the first participant asked “admits they can’t figure out what color dot they’ve got.”
But this? If this had been one of those Knights Who Always Tell The Truth puzzles with Knaves Who Always Lie, then a knight could make that statement, since it’s true right when he says it — regardless of whether it was also true a minute ago, and regardless of whether it’s still going to be true a minute later. It’s true now; is that all that matters, if that’s all he actually says?[/spoiler]
These are “perfect” logicians. They process instantly. When they say “now” I know, they mean it. It’s implied that the “now” in “now I know” statements means that the person speaking those words couldn’t have known before the last person to speak did so, not because they take more time to process or for some other reason. That’s how brain teasers with perfect logicians generally work. The first link in the first reply in this thread states “if a conclusion can be logically deduced, they will do it instantly.”
Gonna skip the spoiler box. This is how it would work if the secret word was “cat”:
Albert has a “c.” When he states “I know the secret word”, Bernard, having a “t”, now knows that the secret word must be “cat.” Bernard didn’t know that before, as with his “t” the word could have been “cat” or “tag.”
Cheryl, having an “a”, knows after Albert’s statement that he must have a “c” or “x” and that the secret word must be “cat” or “max.” It’s not until after Bernard speaks that Cheryl knows the word is “cat.”
But that’s just it: you say it’s implied, but all he actually said is that he knows it now. He doesn’t say that he “couldn’t have known before the last person to speak did so”. He doesn’t say that he wasn’t able to deduce the answer until after Albert spoke up. He only says that he knows it now; which is, near as I can tell, merely true.
You say it implies something more; would a perfect logician be justified in figuring it implies more, or merely that his statement establishes what it says?
When I say it’s implied, I’m not saying anything about a logician being justified in figuring it implies more. I’m saying it’s implied by the person telling the riddle. IME, people reading these riddles understand that when a perfect logician says “now I know”, he hasn’t waited to speak after deducing something and deduces instantly.
I appreciate your faith in me, but I don’t think that’s the reason I’m having trouble. I thnk you’re right, that with enough paper and enough time I could get to the answer; but I’ve not dedicated enough even to get to the point that I realize there’s no solution.
It seems like a variation of this puzzle:
Albert Bernard and Cheryl…
For Albert to know right away, he has to have a unique letter, a letter that’s only in one word.
Unique letters capitalized:
Cat, dOg, HaS, maX, dIm, tag
words with a unique letter:
Cat
dOg
maX
dIm
HaS
plus: tag (no unique letters, it can’t be this one)
of the five:
Bernard has to be holding a letter that in conjunction with Albert’s holding a unique letter tells Bernard what the secret word is.
if “C” then “Cat”; Bernard has to have a or t
but can’t be “a”, due to maX; could be t though
if “O” then “dOg”; Bernard has to have a d or g
but can’t be “d” due to dIm; could be g though
if “X” then “maX”; Bernard has to have an m or a
but can’t be “a”, due to Cat; can’t be m due to dIm:
if “I” then “dIm”; Bernard has to have a d or m
but can’t be “d” due to dOg; can’t be m due to maX;
if “H” then “HaS”; Bernard can’t have the other unique
or he would know as soon as Albert,
so Bernard has “a”; but can’t have “a” either due to Cat
the only words with unique letters Albert can have that would let Bernard also know the word (but only after Albert speaks) are Cat and dOg. Now for Cheryl…
…
of the two:
Cat:
if Albert has C and Bernard has t Cheryl has a
Cheryl could not have known right after Albert
because her a could be the a in maX
dOg:
if Albert has O and Bernard has g, Cheryl has d
Cheryl could not have known right after Albert
because her d could be the d in dIm
Cheryl of course knows her own letter.
if the word is Cat, Cheryl holds an a. Not unique.
Albert’s knowing tells her the word isn’t tag, but at that point it could have been maX or HaS. She only realizes it has to be Cat when Bernard speaks up.
if the word is dOg, Cheryl is holding the g. Not unique. But unique among the words containing unique letters. She would not need to wait for Bernard to speak, she’d know as soon as Albert spoke because she sees her own g.
So “cat” it is.
Um, AHunter3? By your own post:
[spoiler]Albert has O and Bernard has g, Cheryl has d
So Cheryl doesn’t have g, she does need to wait for Bernard to speak, and dog is indeed a possible answer.
[/spoiler]
Dang it…
fixed:
For Albert to know right away, he has to have a unique letter, a letter that’s only in one word.
Unique letters capitalized:
Cat, dOg, HaS, maX, dIm, tag
words with a unique letter:
Cat
dOg
maX
dIm
HaS
plus: tag (no unique letters, it can’t be this one)
of the five:
Bernard has to be holding a letter that in conjunction with Albert’s unique letter tells him what the secret word is.
if “C” then “Cat”; Bernard has to have a or t
but can’t be a, due to maX; could be t
if “O” then “dOg”; Bernard has to have a d or g
but can’t be d due to dIm; could be g
if “X” then “maX”; Bernard has to have an m or a
but can’t be a, due to Cat; can’t be m due to dIm:
if “I” then “dIm”; Bernard has to have a d or m
but can’t be d due to dOg; can’t be m due to maX;
if “H” then “HaS”; Bernard can’t have the other unique
so Bernard has “a”; but can’t have “a” either due to Cat
the only words with unique letters Albert can have that would let Bernard also know the word are Cat and dOg. Now for Cheryl…
…
of the two:
Cat:
if Albert has C and Bernard has t Cheryl has a
Cheryl could not have known right after Albert
because her a could be the a in maX
dOg:
if Albert has O and Bernard has g, Cheryl has d
Cheryl could not have known right after Albert
because her d could be the d in dIm
Cheryl of course knows her own letter.
if the word is Cat, Cheryl holds an a. Not unique.
Albert’s knowing tells her the word isn’t tag, but at that point it could have been maX or HaS. She only realizes it has to be Cat when Bernard speaks up.
if the word is dOg, Cheryl is holding the d. Not unique.
Albert’s knowing tells her the word isn’t tag but it could have been dIm. She only realizes it has to be dOg when Bernard speaks up.
Yeah, it coud be either Cat or dOg.