Question

which three choices are needed to create the figure on the left? (only pieces of the same color may overlap)

a, b, c
b, d, e
a, d, c
c, d, e

Explanation:

Analyze the target figure

The target figure is a \(3 \times 3\) grid with colored squares:

• Row 1: Grey, Blue, Blue
• Row 2: Grey, Grey, Blue
• Row 3: Red, Grey, Blue

Analyze the given pieces

Let's examine the grid coordinates \((row, col)\) from top-left \((1,1)\) to bottom-right \((3,3)\) for each option:

• A:
• Grey at \((2,2)\), \((3,2)\)
• Blue at \((1,3)\), \((3,3)\)
• B:
• Grey at \((2,2)\), \((2,3)\)
• C:
• Red at \((3,1)\)
• Blue at \((1,3)\), \((2,3)\)
• D:
• Grey at \((2,1)\), \((2,2)\)
• Blue at \((1,1)\), \((1,2)\)
• E:
• Red at \((2,1)\)
• Grey at \((2,2)\)
• Blue at \((3,1)\), \((3,2)\)

Test combination A, D, C

Let's overlay pieces A, D, and C:

• Red squares:
• C has Red at \((3,1)\).
• Total Red: \((3,1)\). (Matches target)
• Grey squares:
• A has Grey at \((2,2)\), \((3,2)\).
• D has Grey at \((2,1)\), \((2,2)\).
• Overlap at \((2,2)\) is allowed (both are Grey).
• Total Grey: \((2,1)\), \((2,2)\), \((3,2)\). (Matches target)
• Blue squares:
• A has Blue at \((1,3)\), \((3,3)\).
• C has Blue at \((1,3)\), \((2,3)\).
• D has Blue at \((1,1)\), \((1,2)\).
• Overlap at \((1,3)\) is allowed (both are Blue).
• Total Blue: \((1,1)\), \((1,2)\), \((1,3)\), \((2,3)\), \((3,3)\). (Matches target)

Verify other combinations

• A, B, C: Lacks Grey at \((2,1)\) (only D has Grey at \((2,1)\)).
• B, D, E: Lacks Red at \((3,1)\) (only C has Red at \((3,1)\); E has Red at \((2,1)\)).
• C, D, E: E has Blue at \((3,1)\) and \((3,2)\), which conflicts with the target's Red at \((3,1)\) and Grey at \((3,2)\).

Conclusion

The combination of A, D, and C perfectly reconstructs the target figure.

Answer:

• A, B, C
• B, D, E
• A, D, C (Correct answer)
• C, D, E