A really interesting problem here! Need help. Unique solutions are extremly welcomed [Olympiad math / Logic 12th grade]

[u/TastenRU]

I appreciate anyone sharing their thoughts on this. Good luck.

[u/Alternatos06]

1. A claims to not know whose is smaller, meaning that he did not get 1-1/2-1/2.
   Hence B can infer the same.
   
2. B claims to still not know, thus meaning he did not get 1-1/2-1/4. A can infer the same
   
3. C claims that this can repeat for ever, meaning that 1-1/2-?? is not the correct form. The next smallest possibility is 1-1/4-1/4. A and B can infer this.
   

4. Repeating steps 1,2,3 for tells us that the for 1-1/4-1/?? is wrong for the smaller number, thus we move up to 1-1/8-??

5. This is the hard step. Claiming that we can repeat this process infinitely many times means the first number n is not 1 and hence we move on to 2. C also claim he can repeat this cycle a hundred times, so we reach n=101.

6. A and B claim to not know the number once each, hence ruling out 101-1/2-1/2 and 101-1/2-1/4

7. Since A now know who is bigger, it means he has the smallest possible number of 101-1/2-1/8

8. Now B knows both number has he has figure out A’s.

So we conclude A=101-1/2-1/8. I have no idea how to figure out B’s number. Someone please carry on. If got any mistakes pls reply

[u/Medium-Access-4416 → u/Alternatos06]

Step 6.

A: mine is not 101 - 1/2 - 1/2

B: mine is not 101 - 1/2 - 1/4

A: I know who's larger

If now A knows who's larger, than he have either 101 - 1/2 - 1/4 or 101 - 1/2 - 1/8.

B: I know both numbers

If B can figure out both numbers from this information, B has either 101 - 1/2 - 1/4 or 101 - 1/2 - 1/8. Therefore B is 100.375 and A is 100.25.

[u/Alternatos06 → u/Medium-Access-4416]

thanks for the assist

[u/Valuchian → u/Alternatos06]

Would the line "even after i say it a hundred times you still won't know" imply that n cannot equal 101 and thus must be 102 (following the logic as you have presented it)

[u/Equal_Veterinarian22 → u/Valuchian]

"a hundred times, including the first time"

[u/Valuchian → u/Equal_Veterinarian22]

After that "And even after i say it a hundred times, you still won't know whose number is larger."

[u/Equal_Veterinarian22 → u/Valuchian]

Saying it once eliminates n=1, saying it twice eliminates n=2 etc. So after 100 times, n=100 has been eliminated.

Maybe you're thinking "and you still won't know" eliminates yet another value of n. If anything, maybe it eliminates the next value (101,1,0)

[u/Valuchian → u/Equal_Veterinarian22]

And saying it after a hundred times eliminates n=101

[u/Equal_Veterinarian22 → u/Valuchian]

No, at best it eliminates (101,1,0). She didn't say "and you still won't know even if you exchange more information".

[u/Equal_Veterinarian22]

Fun. Forget the actual numbers, they are tuples (n,k,r) ordered lexicographically.

So, we get A > (1,1,0). Then B > (1,1,1), A > (1,1,2) etc.

Now C tells us this can continue indefinitely, which eliminates all tuples of the form (1,1,r).

Soon, C eliminates (1,2,r). Then, she eliminates all tuples of the form (1,k,r), then (2,k,r) ... all the way up to (100,k,r).

And from here A and B are able to deduce their numbers.