Completing a 3×3 Matrix Item Under a Single Stated Rule
An assessment designer is building a 3×3 matrix item. Each cell holds one figure described by three attributes: shape (triangle, square, pentagon), fill (solid, hatched, empty), and dot count (1, 2, 3). The only stated rule is this: within every row and every column, each value of each attribute appears exactly once.
The grid as drafted, reading left to right:
- Row 1: solid triangle, 1 dot; hatched square, 2 dots; empty pentagon, 3 dots.
- Row 2: empty pentagon, 2 dots; solid triangle, 3 dots; hatched square, 1 dot.
- Row 3: hatched square, 3 dots; empty pentagon, 1 dot; cell missing.
One reviewer says the missing cell is "obviously a triangle with 3 dots," because the main diagonal shows dots 1, then 3, and must complete a pattern. A second reviewer says the item is under-determined and should be dropped. The designer has an hour to decide whether to keep the item and which key to publish.
1. Derive the missing cell using only the stated rule, attribute by attribute.
Read the worked answer
Work down column 3, then cross-check across row 3.
- Shape. Column 3 holds a pentagon (row 1) and a square (row 2), so the missing shape must be a triangle. Row 3 confirms it: square, pentagon, triangle.
- Fill. Column 3 holds empty (row 1) and hatched (row 2), so the missing fill must be solid. Row 3 confirms: hatched, empty, solid.
- Dots. Column 3 holds 3 (row 1) and 1 (row 2), so the missing count is 2. Row 3 confirms: 3, 1, 2.
Missing cell: a solid triangle with 2 dots.
Note that each attribute is forced independently - no attribute requires a guess, and the row and column constraints agree rather than conflict. A conflict between them would be the real sign of a broken item; here there is none.
2. Is the first reviewer's "triangle with 3 dots" defensible? Assess the reasoning, not just the answer.
Read the worked answer
No, and the reasoning is the tell. Three problems:
- Two points do not make a pattern. The diagonal offers 1 and 3; infinitely many rules pass through two values (1, 3, 5; 1, 3, 1; 1, 3, 2). Nothing selects 3.
- It contradicts the stated rule. Row 3 would read 3, 1, 3, repeating a dot count in a row; column 3 would read 3, 1, 3, repeating again. A proposed answer that violates the only stated constraint is not an alternative reading, it is an error.
- It adds a rule the item never announced. Test-takers may invent rules; item writers may not leave an unstated pattern that competes with the stated one and then mark the inference wrong.
The reviewer has done exactly what a careless solver does - spotted a visual coincidence in a Latin-square construction, where diagonals often carry accidental patterns, and treated it as the design principle. The coincidence is why the distractor is tempting, which is legitimate difficulty, but it is not evidence about the key.
3. Answer the second reviewer: is the item under-determined, and should it be kept?
Read the worked answer
Not under-determined. The stated rule fixes each attribute uniquely: column 3 alone forces triangle, solid, 2 dots, and row 3 independently forces the same three values. Because the two constraints agree, the solution is unique rather than arbitrary. Uncertainty in a solver is not indeterminacy in the item.
Keep it, with one design note. The published key should be solid triangle with 2 dots.
The weakness worth logging is different from the one alleged: the answer can be reached from a single column, so the item does not compel the solver to integrate row and column information - it tests one-directional scanning only. A stronger version would break the symmetry so that at least one attribute requires combining a row and a column (for example by staggering which cell is blank across variants). The diagonal coincidence should also be recorded, since it will attract a predictable wrong response - useful as a distractor, harmful only if the key were ever written from the visual pattern rather than the rule.