Two tiles are missing in Panel I. Which one of the options in Panel II is the appropriate choice for the missing tiles?

Let us count the number of filled (black) dots in each $3 \times 3$ tile of Panel I:

• Row 1:

  • Tile $(1, 1)$: $8$ filled dots

  • Tile $(1, 2)$: $1$ filled dot

  • Tile $(1, 3)$: $6$ filled dots

  $$\text{Sum of Row 1} = 8 + 1 + 6 = 15$$

• Row 2:

  • Tile $(2, 1)$: $3$ filled dots

  • Tile $(2, 2)$: $5$ filled dots

  • Tile $(2, 3)$: missing ($x$ filled dots)

• Row 3:

  • Tile $(3, 1)$: $4$ filled dots

  • Tile $(3, 2)$: $9$ filled dots

  • Tile $(3, 3)$: missing ($y$ filled dots)

• Column sums:

  • Column 1: $8 + 3 + 4 = 15$

  • Column 2: $1 + 5 + 9 = 15$

This follows the structure of a standard $3 \times 3$ magic square where every row, column, and diagonal sums to $15$:

Solving for the missing values:

• For Row 2: $3 + 5 + x = 15 \implies x = 7$

• For Row 3: $4 + 9 + y = 15 \implies y = 2$

Checking Column 3: $6 + 7 + 2 = 15$, which is consistent.

Looking at Panel II:

• Option (i) has an upper tile with $7$ filled dots and a lower tile with $2$ filled dots.

Therefore, option (A) corresponding to (i) is the correct choice.