The Symbol Matrix and the Arrow Cycle
Study the 3×3 matrix below. Each cell contains one shape (▲, ■ or ○) and between one and three dots (•). The same rule applies across every row and every column.
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | ▲• | ■•• | ○••• |
| Row 2 | ○•• | ▲••• | ■• |
| Row 3 | ■••• | ○• | ? |
A second item presents a sequence in five described steps: step 1 is an arrow pointing up, shaded; step 2 is an arrow pointing right, shaded; step 3 is an arrow pointing down, unshaded; step 4 is an arrow pointing left, unshaded; step 5 is an arrow pointing up, shaded. The sequence continues by the same rules.
1. Which shape and dot count completes the matrix? Explain the rule and test it against one row and one column.
Read the worked answer
The missing cell is ▲ with two dots. The rule is that every row and every column contains each of the three shapes exactly once and each dot count (1, 2 and 3) exactly once. Test row 3: the shapes are ■ and ○, so the missing shape must be ▲; the dots are 3 and 1, so the missing count must be 2. Test column 3: the shapes are ○ and ■, so the missing shape must be ▲; the dots are 3 and 1, so the missing count must be 2. Both tests agree. A further check is that no row or column repeats a shape-dot combination, which also holds. The answer therefore satisfies two independent constraints, and that agreement is what distinguishes a genuine rule from a coincidental pattern.
2. A candidate proposes that the rule is simply that the number of dots equals the number of sides of the shape. Show why this fails and what that tells you about checking a rule.
Read the worked answer
The 'dots equal sides' rule fails immediately. A circle has no sides, yet ○ appears with one, two and three dots in different cells, and ▲ appears with one dot in one cell and three in another. It is a plausible rule that fits one or two cells and then breaks. The lesson is to test a proposed rule against every row and column, especially the cell you are least sure about, before committing. In this matrix the dots are governed by position, not by shape: the two attributes vary independently. That independence is what makes the item solvable rather than circular, because a rule that derives the dots from the shape would make one of the two constraints redundant.
3. What is step 8 of the arrow sequence, and what assumption does your answer depend on?
Read the worked answer
Step 8 is an arrow pointing left, unshaded. The direction rotates 90° clockwise at each step (up, right, down, left), so the directional pattern repeats every four steps; step 8 therefore has the same direction as step 4, which is left. Shading is shaded for steps 1–2 and 5–6, and unshaded for steps 3–4, so steps 7–8 are unshaded, consistent with step 4. The answer depends on assuming the cycle continues unchanged and that shading changes every two steps rather than being tied to direction. If shading were tied to direction, step 4 and step 8 would still both be left and unshaded, so the answer is robust to that alternative interpretation.