The Nine-Tile Matrix
Nine tiles are arranged in a 3×3 grid. Each tile contains a regular polygon, a number of small dots inside it, and a short marker line from the centre to one edge. The marker can point north, east, south or west. The grid is as follows. Top row: triangle with 1 dot, marker north; square with 2 dots, marker east; pentagon with 3 dots, marker south. Middle row: square with 2 dots, marker east; pentagon with 3 dots, marker south; hexagon with 4 dots, marker west. Bottom row: pentagon with 3 dots, marker south; hexagon with 4 dots, marker west; the bottom-right tile is missing. No tile repeats another, and each row and column follows the same three independent progressions. The options for the missing tile are: (A) heptagon, 5 dots, marker north; (B) heptagon, 6 dots, marker north; (C) hexagon, 5 dots, marker east; (D) heptagon, 5 dots, marker west.
1. Deduce the missing bottom-right tile and state the three independent rules that generate the grid.
Read the worked answer
Work along each row and down each column.
Sides: top row 3, 4, 5; middle row 4, 5, 6; bottom row 5, 6, ?. Each step right adds one side, and each step down adds one side, so the missing polygon has 7 sides (a heptagon).
Dots: top row 1, 2, 3; middle row 2, 3, 4; bottom row 3, 4, ?. The same +1 progression gives 5 dots.
Marker: top row N, E, S; middle row E, S, W; bottom row S, W, ?. Each step right rotates the marker 90° clockwise, and each step down also rotates it 90° clockwise. From W, one clockwise step is N.
So the missing tile is a heptagon with 5 dots and the marker pointing north. That is option A. Option B has too many dots, C has too few sides, and D has the marker pointing west.
2. A colleague proposes the simpler rule 'dots = sides − 2'. Is it consistent with the visible tiles? Does it help identify the missing tile? Explain.
Read the worked answer
It is consistent. Check the visible tiles: 3 sides and 1 dot; 4 and 2; 5 and 3; 4 and 2; 5 and 3; 6 and 4; 5 and 3; 6 and 4. All eight satisfy sides − 2 = dots. The missing heptagon would therefore need 5 dots, which matches the result from the row/column progression.
But it is a derived rule, not an independent one. Sides and dots both increase by exactly 1 at each step right and each step down, so their difference is fixed at the top-left tile: 3 − 1 = 2. The rule adds no information beyond the two progressions, and it cannot choose the marker orientation. On its own it would allow a heptagon with 5 dots and a marker pointing north, east, south or west. It is a useful check, not a substitute for the three generating rules.
3. If the marker rule were removed, would the missing tile still be uniquely determined? Give two candidate tiles that would fit the remaining rules, and explain what this shows about the role of each rule.
Read the worked answer
No. The shape and dot rules still force a heptagon with 5 dots, but they say nothing about the marker. The marker could point north, east, south or west. Two candidates that both fit the remaining rules are option A (heptagon, 5 dots, marker north) and option D (heptagon, 5 dots, marker west). Both have the correct number of sides and dots, and the grid contains no other constraint that would select one orientation over the other.
This shows that the three rules constrain separate attributes: the side progression fixes the polygon, the dot progression fixes the dot count, and the rotation rule fixes the marker. The dots = sides − 2 relationship is not a fourth independent constraint; it is a consequence of the first two progressions. A complete answer has to apply each independent rule to its own attribute.