The Missing Ninth Tile: A 3×3 Shading Puzzle
A weekend puzzle club is restoring a 3-by-3 tile panel from last season's tournament. Eight tiles survive; the bottom-right one is missing and must be recut from a spare sheet. The club's notes give three rules, each already confirmed by the eight surviving tiles. Column 1 tiles are triangles, column 2 tiles are squares, column 3 tiles are pentagons. Row 1 tiles carry one small dot, row 2 carry two, row 3 carry three. Shading follows a checkerboard: no tile shares its shading with the tile to its right or the tile below it, and the top-left tile is shaded.
The surviving panel therefore reads, row by row: shaded triangle with one dot, white square with one dot, shaded pentagon with one dot; white triangle with two dots, shaded square with two dots, white pentagon with two dots; shaded triangle with three dots, white square with three dots, and the gap.
Two spare tiles sit on the bench: an unshaded pentagon with three dots, and a shaded hexagon with three dots. The panel must ship in an hour.
1. Describe the missing tile completely, showing how each of the three rules constrains it.
Read the worked answer
The gap is the third tile of the bottom row, so three independent rules each fix one attribute.
- Shape: column 3 holds pentagons - the top-right and middle-right tiles are both pentagons - so the new tile is a pentagon. A six-sided tile is not in this column's vocabulary.
- Dots: row 3 holds three dots, as the bottom-left and bottom-middle tiles show, so the new tile carries three.
- Shading: the checkerboard rule forbids matching either edge-neighbour. The tile above the gap (middle-right) is white and the tile to its left (bottom-middle) is white, so the gap must be shaded.
Check the parity form of the rule on tiles you already have: shading matches tiles whose row and column have the same parity, which is (1,1), (1,3), (2,2), (3,1) - all shaded - so (3,3) is shaded too.
Answer: a shaded pentagon carrying three dots. Three different routes to the shading answer agree, which is the signal that the reconstruction is right rather than merely plausible.
2. Neither spare tile on the bench satisfies all three rules. Say which rule each one breaks, and explain why 'use the closest match' is the wrong instruction here.
Read the worked answer
Run each spare against the rules:
- Unshaded pentagon, three dots: shape ✓ (pentagon, column 3), dots ✓ (three, row 3), shading ✗. Both edge-neighbours are white, and the checkerboard rule forbids a third white tile in that L-shape.
- Shaded hexagon, three dots: dots ✓, shading ✓ (it differs from both neighbours), shape ✗. Column 3 is the pentagon column, so a hexagon would be the only six-sided tile in the panel.
'Use the closest match' fails because the three rules are not weighted. Each is fixed by the eight surviving tiles, so a tile that satisfies two rules and breaks one is simply wrong - the wrongness is invisible in the attributes it got right. The club must recut a shaded pentagon with three dots.
In item terms this is exactly why distractors are built this way: each wrong option is a near miss on exactly one attribute, so a student who checks only one rule at a time can talk themselves into it.
3. How many shaded tiles does the completed 3-by-3 panel contain, and can that count catch an error in a proposed answer?
Read the worked answer
Five. The shading rule is a checkerboard on a 3-by-3 grid, so the shaded tiles are those whose row and column have the same parity: (1,1), (1,3), (2,2), (3,1) and (3,3). That is four corners plus the centre; the four edge-middle tiles are white.
The count is a cheap audit. If someone proposes an unshaded bottom-right tile, the panel would hold only four shaded tiles, and the bottom-right would sit inside a white L of three tiles - the count catches the error before the shape or dot check is even run. It catches the opposite mistake too: adding a shaded tile to a cell that is not on the shaded pattern would produce six shaded tiles.
A count check works only because the completed panel is forced. If two placements were still open, a total would prove nothing; here each cell is fixed by its row and column, so any total other than five means a rule was misapplied.