Logic Puzzle! (1 Viewer)

[seanieg89]

(Not created by me.)

Cheryl Welcome, Albert and Bernard, to my birthday party, and I thank you for your gifts. To return the favor, as you entered my party, I privately made known to each of you a rational number of the form

,

where n and k are positive integers and r is a non-negative integer; please consider it my gift to each of you. Your numbers are different from each other, and you have received no other information about these numbers or anyone’s knowledge about them beyond what I am now telling you. Let me ask, who of you has the larger number?

Albert I don’t know.

Bernard Neither do I.

Albert Indeed, I still do not know.

Bernard And still neither do I.

Cheryl Well, it is no use to continue that way! I can tell you that no matter how long you continue that back-and-forth, you shall not come to know who has the larger number.

Albert What interesting new information! But alas, I still do not know whose number is larger.

Bernard And still also I do not know.

Albert I continue not to know.

Bernard I regret that I also do not know.

Cheryl Let me say once again that no matter how long you continue truthfully to tell each other in succession that you do not yet know, you will not know who has the larger number.

Albert Well, thank you very much for saving us from that tiresome trouble! But unfortunately, I still do not know who has the larger number.

Bernard And also I remain in ignorance. However shall we come to know?

Cheryl Well, in fact, no matter how long we three continue from now in the pattern we have followed so far—namely, the pattern in which you two state back-and-forth that still you do not yet know whose number is larger and then I tell you yet again that no further amount of that back-and-forth will enable you to know—then still after as much repetition of that pattern as we can stand, you will not know whose number is larger! Furthermore, I could make that same statement a second time, even after now that I have said it to you once, and it would still be true. And a third and fourth as well! Indeed, I could make that same pronouncement a hundred times altogether in succession (counting my first time as amongst the one hundred), and it would be true every time. And furthermore, even after my having said it altogether one hundred times in succession, you would still not know who has the larger number!

Albert Such powerful new information! But I am very sorry to say that still I do not know whose number is larger.

Bernard And also I do not know.

Albert But wait! It suddenly comes upon me after Bernard’s last remark, that finally I know who has the larger number!

Bernard Really? In that case, then I also know, and what is more, I know both of our numbers!

Albert Well, now I also know them!



Question. What numbers did Cheryl give to Albert and Bernard?

 

[GoldyOrNugget]

In the first round A[lbert] doesn't know, so he can't have n=1, k=1, r=0 (otherwise he'd know that he has the minimum possible value and thus that B[ernard] must have a larger value). B then doesn't know, which means that B also doesn't have n=1, k=1, r=0 and he ALSO doesn't have n=1, k=1, r=1. Both A and B now know that both of them don't have n=1, k=1, r=0 and furthermore that B doesn't have n=1, k=1, r=1. A continues not to know, meaning he doesn't have the next smallest possible value n=1, k=1, r=2, and neither does B, meaning bla bla n=1 k=1 r=3. C[heryl] interrupts and asserts that they will never find the solution by this method, meaning that it's not the case that n=1 and k=1. A and B become aware of this.

The hypothetical smallest number now is given by n=1, k=2, r=0. Again, they repeat the process, but C reveals that it's futile, meaning it's not the case that n=1 and k=2. The pattern is that A and B's dialogue iterates through the values of r, and C's interruption increments the k value and resets r. Also, every time C acknowledges that this dialogue+interrupt pattern is futile, the n value is incremented and k and r are reset.

C acknowledges the pattern a hundred times in succession (meaning now n=101, k=1, r=0) and says that both A and B still wouldn't know the answer, meaning neither A nor B have n=101, k=1, r=0. Still A doesn't know the answer (meaning A does not have n=101, k=1, r=1) and subsequently neither does B (meaning B does not have n=101, k=1, r=1 or n=101, k=1, r=2).

A now knows the answer. This means that A has n=101, k=1, r=2 or n=101, k=1, r=3. Either way, he knows that he has the lowest number and thus that B has the highest.

Now B knows the answer, meaning he has n=101, k=1, r=3 or n=101, k=1, r=4. But supposing that he had r=4, he wouldn't be able to decide if A had r=3 or r=2, so he could not possible know the value of both the numbers. So it follows that B must have r=3, and so A must have r=2.

So according to my working (which is bound to have an off-by-one error (or two, or three) somewhere) Albert's number is 100.375 and Bernard's number is 100.4375.

 

[GoldyOrNugget]

Contributing my own logic puzzle:

I have a deck of 20 numbered cards in my hand numbered 1..20. The deck is turned face down so you can't see the numbering on any of the cards, and you don't know what order they're in. I repeat the following steps 20 times in a row:

1. I take the top card off the deck, and lay it on the table in front of you face up.
2. I take the top card off the deck and put it at the bottom of the deck without showing you its number.

Once I complete the demonstration, you notice that I've laid the cards on the table in sequential order from 1 to 20.

What was the original ordering of the cards in my hand?

 

[swagswagyoloswag]

Friend did this dunno if hes Legit though
1 11 2 16 3 12 4 19 5 13 6 17 7 14 8 20 9 15 10 18
is it right?

 

[GoldyOrNugget]

Almost! 19 and 20 should be switched around.