Reading a Figure Sequence Without a Picture
A sequence of five figures is described in words. Each figure has only three features: the number of sides of its outer shape, the number of small shaded squares inside it, and the corner at which a single black dot sits (top, right, bottom or left).
- 3 sides, 1 shaded square, dot at the top
- 4 sides, 3 shaded squares, dot at the right
- 5 sides, 5 shaded squares, dot at the bottom
- 6 sides, 7 shaded squares, dot at the left
- 3 sides, 9 shaded squares, dot at the top
The sixth figure must be chosen from four options:
- A: 4 sides, 11 shaded squares, dot at the right
- B: 4 sides, 11 shaded squares, dot at the bottom
- C: 6 sides, 11 shaded squares, dot at the right
- D: 4 sides, 10 shaded squares, dot at the right
1. Choose the correct option and justify your choice feature by feature, rejecting each wrong option for a specific reason.
Read the worked answer
Feature 1 - sides. The pattern is 3, 4, 5, 6, then back to 3. One full cycle has been completed, so figure 6 should begin the cycle again: 4 sides. This eliminates C (6 sides).
Feature 2 - shaded squares. 1, 3, 5, 7, 9 increases by 2 each time, so figure 6 has 11. This eliminates D (10).
Feature 3 - dot. top → right → bottom → left → top is one quarter-turn clockwise each step. After 'top' comes right. This eliminates B (bottom) - B has the right side count and square count but the dot has been carried on from figure 4 instead of figure 5.
Answer: A.
Notice how each distractor breaks exactly one rule: C breaks sides, D breaks squares, B breaks the dot. Only A satisfies all three simultaneously, which is why a good method checks every feature separately rather than trusting the first one that matches.
2. A pupil says the answer is certain. Give one serious reason why figure 6 is not certain from the evidence given, and describe what extra evidence would settle it.
Read the worked answer
Only one cycle of the side-count has been seen (3, 4, 5, 6, then 3 again at figure 5). A single return to 3 is consistent with two different rules:
- the shape resets to a triangle every four figures (giving 4 sides at figure 6), or
- figure 5 is an exception and the count continues 7, 8… (which would give a 7-sided shape).
Five figures cannot distinguish these. In the same way, 'squares increase by 2' and 'squares are the odd numbers 1, 3, 5, 7, 9, 11' give the same prediction here but would diverge for larger figures if the rule were, say, 'add 2 until 11, then add 4'.
Extra evidence that would settle it: figures 7 and 8 (does the side count go 5, 6 again?), or a stated rule from the puzzle setter.
The judgement being tested: a pattern that fits all the given data is not automatically the rule. Say 'the best-supported answer is A', not 'the answer must be A'.
3. A classmate states: 'Figure 10 will have 20 shaded squares, because 10 × 2 = 20.' Explain what the sequence actually predicts and where the classmate's reasoning slips.
Read the worked answer
The sequence 1, 3, 5, 7, 9 follows the rule 2n − 1 (the odd numbers), not 2n. The classmate has used the wrong formula - the '2' in 'add 2 each time' is a step size, not a multiplier of the figure number.
Figure 10: 2 × 10 − 1 = 19 shaded squares.
Check by extending: figure 5 = 9, figure 6 = 11, 7 = 13, 8 = 15, 9 = 17, 10 = 19.
Why the error is easy to make: 10 × 2 = 20 is only one more than 19, so the mistake looks harmless. But a rule that is off by one at figure 10 will be wrong at every figure, and in a sequence question the formula must be checked at figure 1 (2 × 1 − 1 = 1, correct) and figure 5 (2 × 5 − 1 = 9, correct). Testing a formula against the smallest and largest known terms is the cheapest way to catch this kind of slip.