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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

Labels of the three boxes with closed lids are BB, BR and RR. The boxes have colored balls inside.

One box has only Blue, the second box only Red, and the third—Red and Blue color balls.

The label BB should mean, only Blue balls are in the box. Label RR should mean, only Red balls are in the box and label BR should mean the box has balls of both Blue and Red colors.

The problems is—all the boxes are wrongly labeled.

Three wrongly labeled boxes

Your 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?

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 know the contents of all three boxes.

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

First stage of logical thinking is to understand the information and how it is useful to you.

What do you think is the most valuable information?

Think for a moment.

The color of the balls inside is good information—one box only with Blue balls, one only with Red balls and the third with both Blue and Red balls (you don’t know which box has what color balls).

The second information is—all boxes wrongly labeled.

What does it mean?

This is the point where you are exploring in more details one particular bit of information that seems more important. Sure, there is more useful information hidden in this statement.

Conclusions drawn from the fact of all boxes wrongly labeled: 3 boxes puzzle

Take the RR labeled box for analysis. The label is wrong. It means—this RR labeled box can only have all Blue or both Blue and Red color balls mixed.

What would you expect when you open this box and take out one ball? If the ball is Blue, you would know that the RR labeled box contains either all Blue balls or Blue and Red balls mixed. You won’t be sure. You would face similar uncertainty if the color of ball picked up turns out to be out Red.

As the BB and RR labels are otherwise equivalent, you would face similar uncertainty also for BB labeled box.

Your first conclusion is then,

Conclusion 1: You cannot open either RR or BB labeled box for solving the riddle. It means—you have to open the BR labeled box for solution.

To be clear on this choice, examine the implications of opening the BR labeled box. Sure, its wrong label means—the box can contain only all Red or all Blue color balls. But what would happen after you take out one ball?

If the color of the ball taken out is Red, it would mean—color of all balls inside BR labeled box is Red (the box can contain only Red or only Blue balls).

What would be the color of balls in other two boxes?

Take BB labeled box. It cannot contain all Blue balls, and BR labeled box already contains all Red balls.

Draw your conclusion on the label of box with all Blue balls,

Conclusion 2: The RR labeled box must have all Blue balls in this case.

Conclusion on contents of last BB labeled box is automatic,

Conclusion 3: The BB labeled box has the Blue and Red color balls mixed.

Summarize the results.

Combination 1: Color of ball taken out of BR labeled box Red—the box contains all Red balls, RR labeled box has all Blue balls and BB labeled box has Blue and Red color

What would be the results if color of ball taken out first from BR labeled box turns out to be Blue?

Summary of results for second combination,

Combination 2: Blue ball taken out of BR labeled box, Color of balls in BR labeled box all Blue, BB labeled box has all Red, and RR labeled box has Blue and Red color balls mixed.

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

Following implication diagram should help visualize part of thought chain used for solving the riddle.

Implications flow diagram

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