Three players, Annie, Billy and Katie, each have a natural number written on their foreheads. Each player can only see the foreheads of the other two. The following two things are common knowledge among the players:
- One of the numbers is the sum of the other two.
- One of the numbers is equal to exactly two thirds of one of the other numbers.
Annie, Billy and Katie tell the truth the whole time without exception. Each player is also a perfect logician: anything that they could infer from the information and observations available to them, they instantly will.
The following truthful conversation may or may not take place:
Annie: I don’t know my number.
Billy: I don’t know my number.
Katie: I don’t know my number.
Annie: I now know my number, and it is 150.
Is the above conversation plausible or not? Why?
My answer is NO. There are $2$ cases where one term of addition is $2/3$ from the other term or is $2/3$ from sum. The first player (Annie) doesn't know her number only if she sees two numbers which are in ratio $2:3$ and doesn't know in what case she is. For example Annie sees $300$ and $450$ and can't tell if her number is $150$ or $750$ or she sees $60$ and $9$0 and her number can be $150$ or $30$. Because of this, the second player (Billy) can know his number by computing sum and difference and see which one is in ratio $2:3$ with one of the numbers he sees.
I want to know if what I said is true and if not why. But please I don't want the right answer of the riddle. Thanks