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Three Bag Logic Puzzle
There are three children. These children need rewarding for good behavior. I have six gifts.
I have three chocolate bars.
I have three giant lollypops.
Each child is going to receive two presents and, so they don’t know who is getting what, I put each pair of presents inside a brown paper bag and seal them. Only I know the distribution of the treats, but all the children know there are three of each treat and two treats in each bag.
The first child collects her bag and looks inside to see what presents she received. A curious neutral third party asks this first child a question to determine the breakdown of presents in her bag. The catch is that the child can only respond with the answers: “Yes”, “No” (or “I don’t know”). The child is a perfect logician, and always answers truthfully. What question can the outsider ask to determine the contents of the first gift bag?
What question can you ask the first child to determine the breakdown of presents in her bag?
For the purposes of this exercise, we can assume the chocolate bar and the lollypop have the same mass; This is meant to be a purely logic puzzle and does not rely on physical differences between the bags. There is no trick, gotcha, or tomfoolery in the answer.
Remember, the child can only answer your question with, “Yes”, “No”, or “I don’t know”.
Click button for a hint:
Hint
The key to this puzzle is to realize that there are three possible configurations/states for her bag (two chocolate, two lollypop, or one of each). There are also just three possible responses she can give : “Yes”, “No”, and “I don’t know”.
You need to formulate a question where you can map each of the child’s answers, uniquely, to each of the states.
Click button for the answer:
Solution
There are multiple ways to solve this. I’m not saying my suggestion is any more correct than an answer you might have come up with, but here is my suggestion:
You can ask the first child if any of the other children will receive more chocolate bars than she has.
“Does anyone have more chocolate bars than you?”
Let’s consider the three possible states:
If the first child has two chocolate bars, then she would truthfully answer “No”.
There are only three chocolate bars, and since she has two, nobody could have more.
If the first child has two lollypops, then she would truthfully answer “Yes”.
Both of the other children will have more chocolate bars than she does (one will have two, and the other one).
If the first child has one of each treat, then the only answer she can give is “I don’t know”.
If could be that one of the other children has two chocolate bars (and hence more than she does), or it could be that both children have the same number of each treat. It could be either. There is no answer she can truthfully give other than “I don’t know”.
We've mapped each of the three distinct answers to each of the three distinct states. Problem solved!
It's just a puzzle!
Finally; this was supposed to be a logic puzzle, not a moral conundrum. Please, no comments about why I should have automatically made the bags symmetrical! After all, there might have been external information about the treat preferences of the children that I was aware of before making the bags as they were. This contrived puzzle was meant as a logic exercise to uniquely identify a tri-state system, and is no more artificial a situation than tribes of people who always tell the truth, always lie, or alternate in the telling the truth and lying!
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