Squares and Square Roots | A

Question 4

There are 3 closed boxes—one containing only red balls, the second containing only blue balls and the third containing only green balls. The boxes are labelled RED, BLUE and GREEN such that ‘no’ box has the correct label. We need to find which label goes with which box. How can this be done if we are allowed to open only one box?

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • We have three boxes labelled RED, BLUE, and GREEN.
  • Given condition: Every box is mislabelled (no box has the correct label):
    • Box labelled RED≠Red balls\text{RED} \neq \text{Red balls}
    • Box labelled BLUE≠Blue balls\text{BLUE} \neq \text{Blue balls}
    • Box labelled GREEN≠Green balls\text{GREEN} \neq \text{Green balls}
  • Opening just one box allows us to determine its true contents. By elimination and the mislabelling rule, the contents of the remaining two boxes are uniquely determined.

Step 1 · Choose One Box to Open

Choose and open any box, for example, the box labelled RED, and draw one ball.Diagram 1

Since the box labelled RED\text{RED} cannot contain red balls, the ball drawn must be either Blue or Green.

Step 2 · Case 1: If a Blue Ball is Drawn

If the box labelled RED\text{RED} contains Blue balls:

  • Box labelled BLUE\text{BLUE}: Cannot contain blue balls (already in RED\text{RED}) and cannot contain its own label (blue). Therefore, it must contain Green balls.
  • Box labelled GREEN\text{GREEN}: By elimination, it must contain Red balls.

If RED box has Blue  ⟹  RED=Blue,  BLUE=Green,  GREEN=Red\text{If RED box has Blue} \implies \text{RED} = \text{Blue}, \; \text{BLUE} = \text{Green}, \; \text{GREEN} = \text{Red}

Step 3 · Case 2: If a Green Ball is Drawn

If the box labelled RED\text{RED} contains Green balls:

  • Box labelled BLUE\text{BLUE}: Cannot contain green balls (already in RED\text{RED}) and cannot contain blue balls (its own label). Therefore, it must contain Red balls.
  • Box labelled GREEN\text{GREEN}: By elimination, it must contain Blue balls.

If RED box has Green  ⟹  RED=Green,  BLUE=Red,  GREEN=Blue\text{If RED box has Green} \implies \text{RED} = \text{Green}, \; \text{BLUE} = \text{Red}, \; \text{GREEN} = \text{Blue}

Answer

Open the box labelled RED:

  • If it has Blue balls   ⟹  RED=Blue,  BLUE=Green,  GREEN=Red\implies \text{RED} = \text{Blue}, \; \text{BLUE} = \text{Green}, \; \text{GREEN} = \text{Red}
  • If it has Green balls   ⟹  RED=Green,  BLUE=Red,  GREEN=Blue\implies \text{RED} = \text{Green}, \; \text{BLUE} = \text{Red}, \; \text{GREEN} = \text{Blue}
Common Mistakes
  • Overlooking the "No Correct Label" Rule: Forgetting that every box is strictly mislabelled leads to the false conclusion that opening one box leaves ambiguity for the remaining two.
  • Drawing from More than One Box: The problem allows opening only one box; once its true contents are known, logic and elimination completely resolve the other two boxes without opening them.

More questions in A

Q1

Cut out two identical squares of paper. Draw, label, and cut as follows:

Now place the pieces 5, 6, 7, and 8 around Square 1 to get a square with double the area.

Q2

Halving a Square Using Paper

Cut out a square from a piece of paper. Now make a square whose area is half the area of the first square.

Q3

Will the square having half the sidelength have half the area? Why not? How many such squares will fill the original square?

Fold the square paper inward, as shown, such that the crease lines pass through the midpoints of the sides. PQRSPQRS is the required square with half the area.

Q4

There are 3 closed boxes—one containing only red balls, the second containing only blue balls and the third containing only green balls. The boxes are labelled RED, BLUE and GREEN such that ‘no’ box has the correct label. We need to find which label goes with which box. How can this be done if we are allowed to open only one box?

← Back to Squares and Square Roots