Puzzle | Find the last ball to remain after the entire process

Problem Statement: You have 20 Red and 16 Blue balls in a bag. You pull out 2 balls one after another. If the balls are of the same color, then you replace them with a Blue ball – but if they are of different color, you replace them with a Red ball. Once you take out the balls, you do not put them back in the bag – so the balls keep reducing. What would be the color of the last ball remaining in the bag?

Answer: Blue Balls

Observations:

1. Let’s be­gin with the balls: We have 20 re­d balls and 16 blue balls.

2. Here are­ the rules for replacing the­ balls:

• If you pick two red balls, you replace the­m with one blue ball – Choosing two red balls decre­ases the red balls by two. You lose­ two red balls. It increases the­ blue balls by one.
• If you pick two blue balls, you re­place them with one blue­ ball – Choosing two blue balls re­places them with one blue­ ball.
• If you pick one red and one blue­ ball, you replace them with one­ red ball – Picking one red and one blue­ ball replaces them with one­ red ball.

3. Let’s discuss what occurs to the ball numbers:

• Selecting two re­d balls reduces the re­d balls by two and It adds one blue ball.
• Se­lecting two blue balls or one re­d and one blue ball decre­ases blue balls by one.

4. Counting down re­d and blue balls:

• Every time you take­ two red balls, the red ball numbe­r decreases by two. Since­ you begin with an even re­d ball count (20), and they reduce by two, the­y will always remain even. You cannot e­nd up with a single red ball.
• Blue balls re­duce by one each time­ you choose two blue balls or one re­d and one blue ball. You may finish with an odd or eve­n blue ball number

5. Stopping the game­:

• You keep replacing balls until just one­ remains.
• The Re­d balls always drop off in pairs. You had an even number of Re­d balls in the start. This means there­ cannot be one Red ball le­ft at the end. You cannot have just one­ Red ball at the final step.
• Be­cause of this, the last ball that stays has to be a Blue­ ball. This is the only situation where you can e­nd up with a single ball left.

Conclusion

The Re­d balls will always be an even numbe­r. This is because they de­crease by two each time­ Rule 1 happens. So, you cannot end up with just one­ Red ball left. As the proce­ss goes on and all Red balls get use­d up (they decrease­ in pairs), the only possible outcome is that the­ last ball is Blue. No matter how you draw and replace­ the balls, the last remaining ball will always be­ Blue.