Step 1: Count the filled dots in each tile.
Panel I is a 3 by 3 arrangement of tiles, and each tile is a 3 by 3 grid of dots that are either filled (black) or open (white). Count how many dots are filled in every tile that is already given.
Row 1 tiles have 8, 1, and 6 filled dots.
Row 2 tiles have 3 and 5 filled dots, with the third tile missing.
Row 3 tiles have 4 and 9 filled dots, with the third tile missing.
Step 2: Spot the number pattern.
Write the known counts in a 3 by 3 table:
\[
\begin{matrix} 8 & 1 & 6 \\ 3 & 5 & ? \\ 4 & 9 & ? \end{matrix}
\]
The first row adds up to \(8+1+6=15\), the first column adds up to \(8+3+4=15\), and the middle column adds up to \(1+5+9=15\). Every completed row and column gives the same total of 15. This is exactly the classic 3 by 3 magic square, where 5 sits in the centre and every row, column, and the two diagonals sum to 15.
Step 3: Use the magic square rule to find the missing counts.
For row 2, the missing count must make \(3+5+?=15\), so the missing tile in row 2 needs
\[
? = 15-3-5 = 7
\]
For row 3, the missing count must make \(4+9+?=15\), so the missing tile in row 3 needs
\[
? = 15-4-9 = 2
\]
As a check, the main diagonal gives \(8+5+2=15\), which confirms the pattern is consistent. So the two missing tiles must have 7 filled dots and 2 filled dots, in that order from top to bottom.
Step 4: Match this to the options.
Option (i) shows a pair of tiles where the top tile has 7 filled dots and the bottom tile has 2 filled dots, matching the counts found above exactly.
Option (ii) has a top tile with only 6 filled dots, which does not equal the required 7, so it is ruled out.
Option (iii) and option (iv) both have a bottom tile with 3 or 4 filled dots instead of 2, so both are ruled out.
Final Answer:
Only option (i) gives the correct pair of tiles that keeps every row and column of the magic square summing to 15.
\[ \boxed{\text{(i)}} \]
