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: Count the filled dots in each known tile.
Label the 3 by 3 layout of tiles by (row, column), with row 1 at the top and column 1 at the left. Counting the black dots in each of the seven given tiles: tile (1,1) has \(8\) filled dots, tile (1,2) has \(1\) filled dot, tile (1,3) has \(6\) filled dots, tile (2,1) has \(3\) filled dots, tile (2,2) has \(5\) filled dots, tile (3,1) has \(4\) filled dots, and tile (3,2) has \(9\) filled dots (fully shaded). The two missing tiles sit at (2,3) and (3,3).
Step 2: Find the rule behind the counts.
Write the seven known counts in their grid positions:
row 1: \(8,\ 1,\ 6\)
row 2: \(3,\ 5,\ x\)
row 3: \(4,\ 9,\ y\)
Every complete row and column here adds up to \(15\): row 1 gives \(8+1+6=15\), column 1 gives \(8+3+4=15\), and column 2 gives \(1+5+9=15\). This is a 3 by 3 magic square with magic constant \(15\).
So row 2 needs \(3+5+x=15\), giving \(x=7\): tile (2,3) has \(7\) filled dots. Row 3 needs \(4+9+y=15\), giving \(y=2\): tile (3,3) has \(2\) filled dots.
Step 3: Pin down exactly which dots are filled, not only how many.
Arranging the known tiles by increasing count, \(1,3,4,5,6,8,9\), each tile's filled dots always include every dot from the previous tile in this order, plus one more. The dots are added in this fixed order: the bottom right corner first, then the rest of the rightmost column from bottom to top, then the middle column from bottom to top, then the leftmost column from bottom to top.
Following this order, the tile with \(7\) filled dots (tile (2,3)) fills the whole rightmost column, the whole middle column, and the bottom cell of the leftmost column, leaving only the middle and top cells of the leftmost column empty.
The tile with \(2\) filled dots (tile (3,3)) fills only the bottom two cells of the rightmost column, leaving the top cell of that column and the whole middle and left columns empty.
Step 4: Match this against the four options.
Option (i) shows an upper tile with exactly \(7\) dots filled in this pattern and a lower tile with exactly \(2\) dots filled in this pattern, matching tile (2,3) followed by tile (3,3) exactly.
Option (ii) has only \(6\) filled dots in its upper tile, which breaks the row-2 sum of \(15\). Option (iii) has the correct upper tile with \(7\) dots, but its lower tile fills all \(3\) cells of the rightmost column instead of just the bottom \(2\), so it does not match tile (3,3). Option (iv) again has only \(6\) filled dots in its upper tile.
Final Answer:
Only option (i) reproduces both missing tiles exactly.
\[
\boxed{\text{(i)}}
\]
