The Matrix With a Dead Column
An incomplete 3 x 3 matrix, described cell by cell.
Row 1: a small grey circle at the left edge of its cell, a medium grey circle centred, a large grey circle at the right edge.
Row 2: a small grey square at the right edge, a medium grey square centred, a large grey square at the left edge.
Row 3: a small grey triangle centred, a medium grey triangle centred, and an empty cell at the bottom right.
Answer choices for the empty cell: (A) large triangle, centred; (B) large triangle, at the right edge; (C) medium triangle, at the right edge; (D) large triangle, at the left edge.
1. Sort the three attributes - shape, size, position - into 'rule-governed' and 'undetermined', using only the completed cells.
Read the worked answer
Shape: rule-governed. Constant within each row (circles, squares, triangles). Row 3 needs a triangle.
Size: rule-governed. Both complete rows run small, medium, large strictly left to right. Two independent rows agreeing is real support. Row 3 needs a large figure.
Position: undetermined. Three problems:
- Rows 1 and 2 give opposite sequences (left-centre-right versus right-centre-left), so no single left-to-right rule exists.
- Row 3 shows centre, centre: no variation at all, so the row supplies no information about how position should change.
- Columns give nothing either - column 1 runs left, right, centre; column 2 is centre throughout.
Any position claim must therefore be invented rather than read off the matrix.
2. Choose between (A) and (B) and state the principle behind the choice.
Read the worked answer
Choose (A), large triangle centred.
(A) uses only the two attributes that are actually constrained - size and shape - and reports position in the only way row 3 exhibits it, centred. It adds nothing.
(B) imports row 1's left-to-right sequence. Row 2 already contradicts that sequence by running the other way, so the import is not a pattern but a preference for the first row you happened to read.
The principle: when one attribute is undetermined, decide the item on the attributes that are determined, and let the undetermined attribute take the value that requires no unseen rule. An answer choice that is right on two attributes and neutral on the third beats one that gambles on the third. Note that this makes (A) correct even though it is not 'fully predicted' - a partially determined cell still has a best answer.
3. A classmate argues for (B) by claiming the rows alternate direction: left-to-right, then right-to-left, so row 3 runs left-to-right. What in the matrix defeats this argument?
Read the worked answer
Row 3's own first two cells defeat it.
Under an alternation rule, row 3 runs left-to-right, so the small triangle would sit at the left edge and the medium one in the centre. The matrix shows small centred and medium centred. The proposed rule mispredicts a cell that is already printed, so it is refuted by the data before it ever reaches the empty cell.
The alternation idea is attractive because it uses all three rows and produces a definite answer - but a rule that fits two rows and contradicts the third is worse than no position rule at all. The only position claim consistent with row 3 is 'centred, centred, centred', which again supports (A) or, at most, position-neutral responses.