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Three boxes riddle - the age old logic puzzle

3 boxes puzzle solution by elementary logic analysis

3 boxes puzzle solution by logic analysis and reasoning

3 boxes puzzle: You have 3 boxes containing colored balls. All 3 boxes are wrongly labeled. Open only one box, pick one ball and correctly label the boxes.

The puzzle

Three boxes with closed lids are labeled BB, BR and RR. The boxes have colored balls inside.

One box has only blue, second box only red and the third—red and blue mixed colored balls.

The label BB says, only blue balls are in the box. Label RR says, only red balls are in the box and label BR says, the box has balls of both blue and red colors.

The problems is—all the boxes are wrongly labeled.

And the job is to open exactly one box, take out only one ball and tell the color of balls in each of the three boxes.

Time limit: 10 minutes.


You just cannot open a box randomly and find the answer, can you?

You have to calmly analyze first the nature of the problem in more details. And then after clearing up the doubts and questions that pop up in your mind, you will know exactly which box to open to correctly tell the contents of the box.

Quick Solution to the 3 boxes puzzle: An exercise in elementary logic analysis and reasoning

First stage of logical thinking with reason is, to clearly understand what are given and useful to you for solving the problem.

What do you think is the most valuable information given?

Think for a moment.

The colors of the balls inside surely is good information—one box only with blue balls, one only with red balls and the third with blue and red mixed color.

The second given information is—all boxes are wrongly labeled.

What does this imply?

This is the point where you are exploring in more details one particular bit of information given that seems more important than the other bit of information. Surely there is more useful information hidden in this given information. We call this as Rich information.

The precise conclusions that can be drawn from the fact that all boxes are wrongly labeled: 3 boxes puzzle

Take the RR labeled box to analyze. The label is wrong. This implies, this RR labeled box can only have all blue or blue red mixed color balls.

Just like this result examine the implications of BB labeled box—it implies, the box can contain only all red or blue red mixed color balls.

Together the two implies, blue red mixed color balls must be inside one of these two boxes. The other box will have either all blue or all red balls.

What could be the result of opening any one of these two?

Say, if you open BB box and the ball taken out turns out to be red what will be the implication? It would simply mean the color of balls inside the BB labeled box is either all red or blue red mixed—you won't know for sure. A failure.

Obviously, by now you can see that if you open the RR labelled box, the final result will be the exactly similar uncertainty.

Conclusion: RR and BB labeled boxes you can't open first.

At this point itself you can confidently decide to open the BR labeled box, knowing surely that this would give you the solution.

You have your solution of opening the BR labeled box first. But you have not examined yet in details what exactly the results of opening the BR labeled box could be.

Implications for opening the BR labeled box: 3 boxes puzzle

If the ball taken out of BR labeled box turns to be blue, it implies with certainty that all balls inside are blue. This further implies that the BB labeled box must be having only all red balls. Why? because the RR labeled box cannot have all red balls, that's why. Out of two possibilities of ball color in RR labeled box, BB and RR are not possible. So it can contain only blue red mix colored balls.

Exactly in the same way, if the BR labeled box contains all red balls, the RR labeled box must have all blue balls and the BB labeled box blue red mix colored balls.

You have your solution nice and clear: you will open the BR labeled box first and take out one ball from it. If it is blue, BR box has all blue, BB box has all red and RR box has mixed color balls.

Exactly similar and reverse result if the ball taken out is red.

You have solved this logic puzzle using the technique of raising questions that are important at the moment and examining the implications of the result of taking the action referred to in the question.

By analyzing the implications together you have got your solution.

This is not only directed logic analysis, but also the use of the question, analysis, answer or QAA technique.

These implications are shown in the following diagram. It helps to visualize the thought chain you have used to reach your solution.

implication flow diagram

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