Nine Tiles for a Corridor Display
A workshop is assembling a 3-by-3 display of nine square tiles for a school corridor. Each tile shows one regular polygon with a number of small dots inside it, and the arrangement follows two rules that hold across every row and every column. Row 1 shows a triangle with 1 dot, a square with 1 dot, and a pentagon with 1 dot. Row 2 shows a triangle with 2 dots, a square with 2 dots, and a pentagon with 2 dots. Row 3 shows a triangle with 3 dots and a square with 3 dots; the ninth tile, in the bottom-right corner, was broken in transit. Blank tiles are on hand for triangles, squares, pentagons and hexagons, with room for 1 to 4 dots. Two volunteers disagree about the replacement. One wants a hexagon with 3 dots, arguing that the shape must keep gaining a side. The other wants a pentagon with 4 dots, arguing that the dot count must keep climbing. The display must be hung by Friday, and only one tile will be made.
1. Which tile completes the display? State the answer, name the two rules, and show how each rule is confirmed by more than one line of tiles.
Read the worked answer
The tile is a pentagon with 3 dots.
Read across any row and the sides run 3, 4, 5 - triangle, square, pentagon - so horizontal position fixes the shape: column 1 is always a triangle, column 2 always a square, column 3 always a pentagon. Read down any column and the dots run 1, 2, 3, so vertical position fixes the count: row 3 always carries 3 dots.
The missing tile sits in column 3 and row 3, so it must be a pentagon carrying 3 dots.
Each rule has two independent witnesses. The polygon rule is confirmed by rows 1 and 2 in full, plus the two surviving tiles of row 3. The dot rule is confirmed by columns 1 and 2 in full, plus the two surviving tiles of column 3. Because each attribute is controlled by exactly one direction, the two rules can be combined without interference - which is what makes the answer forced rather than merely plausible.
2. Explain why 'hexagon with 3 dots' and 'pentagon with 4 dots' each look convincing, and identify which surviving tile each one contradicts.
Read the worked answer
Both wrong tiles come from extending one neighbour along the wrong axis.
- Hexagon with 3 dots. It takes the pentagon at row 2, column 3 and adds a side on the theory that shapes grow as you move down. Column 1 contradicts this: row 1 has a triangle and row 3 has a triangle, so moving down a column never changes the shape.
- Pentagon with 4 dots. It takes the square at row 3, column 2 and adds a dot on the theory that dot counts grow as you move right. Row 1 contradicts this: the count stays at 1 from triangle to square to pentagon.
In each case a single neighbour supplies a step that looks like progress but is really coincidence. The reliable habit is to test a suspected rule in the perpendicular direction before committing; a rule that survives only one line of tiles is a description of that line, not of the display.
3. Suppose the bottom-left tile of row 3 had also been lost, so only the first two tiles of each row survived. Would the answer still be forced? Justify your conclusion.
Read the worked answer
Yes. Each attribute needs two witnesses in its own line, and both survive.
- Shape comes from column 3, where rows 1 and 2 still show a pentagon. Whatever row 3 contains in columns 1 and 2, column 3 must keep its pentagon.
- Dots come from row 3, where the triangle and the square both still show 3 dots. That fixes the count for the whole row regardless of which column the missing tile occupies.
So the answer remains a pentagon with 3 dots. What the extra loss would destroy is the row shape sequence (triangle, square, pentagon), which was never a rule anyway - it merely echoed the column rule. This is the useful lesson from the item: identify which line each attribute depends on, and check whether that line retains enough evidence after any damage.