The Shifting Three-by-Three Grid
A 3x3 grid is filled left to right, top to bottom. Each cell holds one shape, one dot count and one shading. Row 1: hollow triangle with 1 dot; solid circle with 2 dots; hollow square with 3 dots. Row 2: solid circle with 2 dots; hollow square with 3 dots; solid triangle with 4 dots. Row 3: hollow square with 3 dots; solid triangle with 4 dots; final cell blank. Shapes are drawn from a fixed set of three, dot counts are whole numbers, shading is either solid or hollow. Only these three attributes vary, and each follows a single rule across the whole grid, with no exceptions.
1. State the rule governing each attribute and give the full contents of the blank cell.
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Shape: the sequence shifts one place left each row. Row 1: T, C, S; Row 2: C, S, T; Row 3: S, T, C. Dots: +1 along each row and +1 down each column. Row 3 runs 3, 4, 5; column 3 runs 3, 4, 5. Shading: solid where the dot count is even (2 and 4 are solid; 1 and 3 are hollow). Five dots is odd, so hollow.
Blank cell: hollow circle with 5 dots.
Cross-check the fit: every solid cell in the grid has an even dot count and every hollow cell an odd one, with no exceptions, so the parity rule is not contradicted anywhere.
2. A colleague proposes instead that 'shading alternates along each row, starting hollow in rows 1 and 3 and solid in row 2'. Does this fit the visible grid, and does it change the blank cell?
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It fits the visible cells: Row 1 is hollow, solid, hollow; Row 2 solid, hollow, solid; Row 3 hollow, solid, so the missing cell is hollow.
So the two rules are indistinguishable on the evidence, but they agree on the answer, which is hollow. That is the useful conclusion: the blank cell is robust to the ambiguity, so the ambiguity does not need resolving before answering.
The disparity rule and the parity rule would only diverge on a grid extended to more columns, so note the limitation rather than guessing.
3. The pattern is extended to a fourth row. What is its first cell, and what assumption does your answer rely on?
Read the worked answer
Shape: the left-shift continues, so Row 4 returns to T, C, S; first shape is triangle. Dots: row and column both increment by 1, so Row 4 runs 4, 5, 6; first count is 4. Shading: 4 is even, so solid.
Answer: solid triangle with 4 dots.
Assumption: that the dot rule is additive (row index + column index - 1, which equals the observed values 1-5) rather than a rule bounded at 5. If the designer intended dot counts to cap or cycle at some value, the fourth row would differ. The shape and shading rules extrapolate cleanly; the dot rule is the one carrying the extrapolation risk.