Panel I shows a 3 by 3 layout of tiles, where each tile is itself a smaller 3 by 3 grid of filled and open dots. The two tiles in the bottom right corner of Panel I are missing. Panel II shows four candidate pairs of tiles, labelled (i) to (iv), that could fill this missing space.
Two tiles are missing in Panel I. Which one of the options in Panel II is the appropriate choice for the missing tiles?
Show Hint
Compare the tiles by how many dots are filled and in what order new dots get added going from a smaller filled count to a larger one, then use the row and column totals as a cross check.
Step 1: Look for a growth pattern across the known tiles.
Arrange the $7$ given tiles by how many dots they have filled: $1, 3, 4, 5, 6, 8, 9$. Comparing each one to the next smaller one, every tile keeps all the filled dots of the smaller tile and adds exactly one more dot in a fixed spot. Tracing this through the full set shows dots get added in this order: bottom right corner, then upward through the rest of the right column, then upward through the middle column, then upward through the left column.
Step 2: Locate where the sequence stands right before and after the missing tiles.
The tile with $6$ dots filled (columns 2 and 3 completely shaded) is the last stop before the left column starts filling in. The very next stage in the sequence adds the bottom cell of the left column, giving $7$ filled dots in total, which is what the missing tile in position (2,3) should show. One stage after that adds the middle cell of the left column, giving $8$, which matches the known tile with $8$ filled dots and only its top left corner empty. So the tile between $6$ and $8$ in the sequence, with $7$ dots, is the correct target for (2,3).
Step 3: Work out the other missing tile the same way.
Near the start of the sequence, the tile with just $1$ dot filled has only the bottom right corner shaded. The next stage fills in the rest of the right column, giving $3$ filled dots, which matches the known tile with $3$ filled dots. The stage with $2$ filled dots sits between these two, and it must fill the bottom right corner plus the cell directly above it, leaving the top cell of the right column, and the whole middle and left columns, empty. This is the pattern needed for the missing tile at (3,3).
Step 4: Check the totals as a cross reference.
Writing the filled dot counts in their tile positions gives a 3 by 3 magic square with every row and column adding to $15$: $8,1,6$ on top, $3,5,7$ in the middle, and $4,9,2$ on the bottom. This agrees with $7$ and $2$ found above.
Step 5: Match against the options.
Option (i) is the only one whose upper tile shows exactly $7$ dots in the right spots (both right and middle columns full, plus the bottom left cell) and whose lower tile shows exactly $2$ dots in the right spots (only the bottom two cells of the right column). Options (ii) and (iv) fall short with only $6$ dots in the upper tile, and option (iii) gets the upper tile right but shades all $3$ cells of the right column in the lower tile instead of just $2$.
\[ \boxed{\text{(i)}} \]
