The 4×4 Grid With a 2×2 Hole: Reading Wrong Answers as Rules
Setting. A curriculum coordinator is vetting a locally written practice set for a fall nonverbal screening of 34 kindergartners that her school runs itself. Item 4 is a 4-by-4 grid of squares. The top-left square is blue. Colours alternate along every row and every column, so no two squares that share an edge are the same colour. The bottom-right 2-by-2 block has been cut out and replaced by one blank outlined square, and the child picks the piece that fills the hole. The choices are: A, one blue square; B, a 2-by-2 block, blue at top-left and bottom-right, yellow at top-right and bottom-left; C, a 2-by-2 block that is all yellow; D, a 2-by-2 block, yellow at top-left and bottom-right, blue elsewhere. Of seven children who tried the item, four chose B, two chose D, one chose C. The coordinator has 90 minutes of meeting time and must decide whether to keep, revise, or retire the item before the set goes to the printer.
1. Which choice is correct, and what does each wrong choice reveal about the rule the child was actually using?
Read the worked answer
Correct choice: B.
The grid is a checkerboard, so colour is fixed by the parity of the row and column. Number rows and columns 1–4 from the top-left, which is blue; a square is blue when row + column is even and yellow when it is odd. The hole covers rows 3–4 and columns 3–4:
- (3,3) = 6, even → blue
- (3,4) = 7, odd → yellow
- (4,3) = 7, odd → yellow
- (4,4) = 8, even → blue
So the patch needs blue at top-left and bottom-right, yellow elsewhere - B.
What the errors show:
- D (2 children) is the mirror of B. These children picked up 'alternating' as a global rule but never anchored it to the blue top-left corner, so either diagonal looked equally good to them. This is a partial rule, not a guess.
- C (1 child) matches only a local, surface cue: the squares nearest the hole are yellow. That is colour-matching, not rule-following.
- A (0 children here) ignores the size of the missing region entirely, which is why nobody chose it - the item gives no support for a one-square answer.
2. The coordinator is tempted to drop D because it is 'too close' to B. Argue for or against keeping it, and say whether the stem needs changing.
Read the worked answer
Keep D, change nothing in the stem.
D is the most informative option in the item. It separates a child who has the alternating rule from a child who also anchors that rule to the given corner - exactly the distinction a pattern-completion item is built to make. Removing D would collapse the item into a two-way choice between an obviously wrong colour and an obviously wrong size, and nearly every child would score it correctly regardless of reasoning.
The evidence is a textbook near-miss pattern: 4 correct, 2 on the mirror, 1 on surface colour, 0 on the size error. That spread is coherent - each distractor captured a different, plausible misreading, and none was dead weight.
What the coordinator should actually fix is placement and interpretation. Put the item in the middle of the set, not first. And treat 7 children as a hypothesis, not a statistic: there is no stable discrimination index at n = 7. The practical step is to run the same item again with the next group and confirm that the B/D split persists before drawing any conclusion about the children or the item.