Arthur and Bert each writes down a positive integer on a piece of paper and then shows it to Charles. Charles then writes two numbers on a blackboard, visible to Arthur and Bert: one of them is the sum of Arthur's and Bert's numbers, and the other is a random number.
After this Charles asks Arthur if he knows Bert's number. If Arthur says he doesn't know, then he asks Bert if he knows Arthur's number. If Bert says he doesn't know, Charles continues with Arthur, then if necessary with Bert and so on... until he gets a positive answer.
When will Charles get a positive answer?
There are 2 random numbers on the board. 1 of them is the sum of Arthur
and Bert's numbers and 1 of them is random. Since both Arthur and Bert
know their own numbers, they can subtract their own number from both of
the numbers on the board. There is no rule against wrong answers, so
both Arthur and Bert just need to try at most twice. So far we have:
Arthur - Says I don't know
Bert - Says I don't know
Arthur - Subracts own number from a number on the board
If right, end sequence
If wrong, continue:
Bert - Can either choose a number on board and do the same as Arthur,
or can guess Arthur's number by subtracting Arthur's guess from a
number on the board.
If right, end sequence
If wrong, continue:
Arthur - Subracts own number from the other number on the board ( the
first time he got it wrong so he knows that the number that he
subracted his own number from before is wrong)
There is no possibility of Arthur getting Bert's number wrong: He
tested both numbers on the board, and if the first one was negative,
this one would be positive.
There are multiple answers to this problem, based solely on whether Arthur or Bert got their first guess right or not.
The answers are:
If Arthur guesses right the first time after saying "I don't know", the
answer is: Charles will get a positive answer on the first try after
Bert says "I don't know" - 50% chance
If Arthur guesses wrong and Bert guesses correct, Charles will get a
positive answer on the second try after Bert says "I don't know" - 25%
chance
If Arthur and Bert both guess wrong the first time, then Charles will
get a positive answer on the third try after Bert says "I don't know" -
12.5% chance, but since Arthur CAN NOT guess wrong on this try, the
sequence ends.
Numerically: 1 try, 50%
2 tries, 25%
3 tries, 12.5%
4 tries, 0 %
I don't know if you want a single answer or not. But if you're asking
when Charles will DEFINITELY get the right answer, no chance involved,
I would say ~the 3rd time he asked Arthur~.
The 2nd time he asked Arthur, Charles would have a 50% chance of
getting a positive answer. The 2nd time Charles asked Bert, he would
have a 25% chance of getting a positiv answer. The 3rd time he asked
Arthur, Charles would have a 100% chance of getting a positive answer,
because Arthur had already eliminated all other wrong posssibilities.
This kind of thinking comes from a strong logical/mathematical intelligence + programming skills. ;)
Hmmm....
