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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:

  1. One of the numbers is the sum of the other two.
  2. 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

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  • $\begingroup$ What if Annie saw 100 and 50? $\endgroup$ – CSCFCEM Aug 25 '15 at 14:28
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    $\begingroup$ 100 and 50 ar not in ratio 2:3 and compute sum and difference: sum = 150 dif = 50, but only sum is in ratio 2:3 with one of the numbers she sees which is 100. Annie can say her number is 150. $\endgroup$ – Jorj Aug 25 '15 at 14:33
  • $\begingroup$ Ah yes, she would not have said her first sentence in that case. I missed that part. $\endgroup$ – CSCFCEM Aug 25 '15 at 14:38
  • $\begingroup$ I would say you are spot on right, but haven't quite explained the final sentence as clearly as you might in terms of what Billy knows and deduces. You might conclude by saying that if $A's$ first statement is true then $B's$ alleged first statement is false $\endgroup$ – Mark Bennet Aug 25 '15 at 14:46
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Your logic is fine. The common knowledge says that the numbers are in either the ratio $1:2:3 \text{ or } 2:3:5$. As you say, Annie must see two numbers in the ratio $2:3$ to make her statement. Then Billy will not see numbers in the ratio $2:3$ and (by the same argument we made about Annie) knows his number without referring to Annie's statement at all.

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It seems that if she saw 100 and 50 then she would be able to shout out this answer with confidence @CSCFCEM said in the comments. This is because 100:150 is a 2:3 ratio, the initial pieces of 'common knowledge' dont specify which number each statement refers to, so the ratio of 50:100 is irrelevant in this case

EDIT: it occurs to me that in this instance there is no real need for the initial logic argument, there may be other combinations where this makes conversation is necessary in order for her to determine the results

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