The Pattern-Grid Deduction
A 3×3 grid follows two simultaneous rules, one applied across rows and one down columns.
Row 1: shapes are circle, triangle, square. Row 2: shapes are triangle, square, circle. Row 3: shapes are square, circle, and one unknown cell.
Separately, a shading rule operates: each row and each column contains exactly one shaded shape, one half-shaded shape, and one unshaded shape. In the current grid, Row 1 is shaded, half-shaded, unshaded (left to right). Row 2 is unshaded, shaded, half-shaded. Row 3 is half-shaded, unshaded, and the unknown cell's shading is not yet given.
1. Determine the shape and shading of the unknown cell, showing how each rule constrains the answer.
Read the worked answer
Shape rule: each row and column contains circle, triangle, square exactly once. Row 3 already has square and circle, so the missing shape is a triangle. Column 3 check: Row 1 square, Row 2 circle, so Row 3 must be triangle - consistent.
Shading rule: each row and column contains shaded, half-shaded, unshaded exactly once. Row 3 already has half-shaded and unshaded, so the missing shading is shaded. Column 3 check: Row 1 unshaded, Row 2 half-shaded, so Row 3 must be shaded - consistent.
The unknown cell is a shaded triangle. Both rules agree independently, which is what makes the answer forced rather than guessed.
2. A classmate claims the answer could also be a half-shaded triangle. Using the column constraint, explain why that claim fails.
Read the worked answer
Column 3 already contains an unshaded shape (Row 1) and a half-shaded shape (Row 2). If Row 3 were half-shaded, Column 3 would contain two half-shaded shapes and no shaded shape, violating the 'exactly one of each shading per column' rule.
So the half-shaded triangle is impossible regardless of the row reasoning. The row constraint alone already forces 'shaded', and the column constraint independently rules out 'half-shaded'. A claim that survives only one rule but breaks the other is invalid; both rules must hold simultaneously.