The Missing Cell in the Pattern Grid
A three-by-three grid of figures is described cell by cell; rows are numbered 1 to 3 top to bottom and columns 1 to 3 left to right. Each cell holds one arrow and some small dots.
Arrows: row 1 - north, north-east, east. Row 2 - south-east, south, south-west. Row 3 - west, north-west, cell (3,3) is blank.
Dots: row 1 - 1, 2, 3. Row 2 - 2, 3, 4. Row 3 - 3, 4, blank.
Fill: row 1 - outline, filled, outline. Row 2 - filled, outline, filled. Row 3 - outline, filled, blank.
Four answer options are offered for cell (3,3):
(a) north arrow, filled, 4 dots (b) north arrow, outline, 5 dots (c) north-west arrow, outline, 5 dots (d) north-east arrow, filled, 5 dots
1. Determine the rule for each of the three attributes and state what must occupy cell (3,3).
Read the worked answer
Arrows - a single sequence read left to right, top to bottom, turning 45 degrees clockwise each step:
N, NE, E, SE, S, SW, W, NW, ... → the ninth term continues the turn to north.
Dots - the count increases by 1 along each row and by 1 down each column, which is the same as dots = row + column − 1:
- (1,1) = 1, (1,3) = 3, (3,1) = 3, (3,2) = 4 → (3,3) = 5.
Fill - a checkerboard alternation: every cell differs from its horizontal and vertical neighbours.
- (3,2) is filled, so (3,3) must be outline; this matches (2,3) filled and (1,3) outline.
Cell (3,3): north arrow, outline, 5 dots → option (b).
2. Eliminate each of the three wrong options by naming the specific rule each one breaks.
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- (a) north, filled, 4 dots - arrow correct, but breaks two rules: the fill rule (its left neighbour (3,2) is filled, and vertical neighbour (2,3) is filled, so the cell must be outline) and the dot rule (the row is 3, 4, so the next is 5, and the column is 3, 4, so the next is 5 - not 4).
- (c) north-west, outline, 5 dots - dots and fill correct, but the arrow repeats (3,2) instead of continuing the 45-degree clockwise turn. Note the trap: NW is the previous term in the sequence, so a candidate working right-to-left lands here.
- (d) north-east, filled, 5 dots - dots correct, but NE is the arrow two steps back in the sequence (it is (1,2)), and the fill breaks the checkerboard.
Only (b) satisfies the arrow sequence, the dot count and the fill alternation simultaneously.
3. A second candidate agrees the answer is north and 5 dots but insists the cell must be filled, arguing that the two outline cells on the main diagonal force the third to be filled as well. Assess this argument and say what would have made the puzzle genuinely ambiguous.
Read the worked answer
The argument is factually wrong about the diagonal. The main diagonal is (1,1) outline, (2,2) outline, (3,3) ?. Two outline cells in a row do not force a change; if the diagonal rule were "all three the same", the third would be outline, not filled. The candidate has also reversed the alternation - the checkerboard rule already fixes (3,3) as outline from two independent directions (its left neighbour and its upper neighbour are both filled).
The real point: two of the three attributes are determined by more than one line of reasoning, so (b) is not a guess. The answer survives if you delete any one row or column.
What would have made it genuinely ambiguous:
- Only two rows and two columns of data, so the sequence could be read in more than one direction.
- A dot pattern with one row inconsistent with the column rule, leaving the arithmetic unresolved.
- A fill pattern with an even number of cells per line, where "alternate" and "opposite ends match" give different answers.
When a pattern is under-determined, the correct move is to say so and state the assumption - not to choose the option that feels most regular.