When the Differences Grow: Frames 1 Through 7
A practice item shows six frames and asks for the seventh. Each frame contains a number of small squares arranged in a single row, and the row is enclosed by a regular polygon. Frame 1 holds 3 squares inside a triangle; frame 2 holds 4 squares inside a square; frame 3 holds 6 squares inside a pentagon; frame 4 holds 9 squares inside a hexagon; frame 5 holds 13 squares inside a heptagon; frame 6 holds 18 squares inside an octagon.
The options are A 22 squares inside a nonagon; B 23 squares inside a nonagon; C 24 squares inside a nonagon; D 24 squares inside an octagon; E 27 squares inside a decagon.
A student, Leo, marks B. He writes: "The gaps are 1, 2, 3, 4, 5, so the next gap is 5 again and 18 plus 5 is 23. The outside shape gains one side each time, so it becomes a nine-sided shape."
1. Give the correct option and show the table that produces it.
Read the worked answer
Answer: C, 24 squares inside a nonagon.
Build the difference column rather than eyeballing it:
| Frame | Squares | First difference |
|---|---|---|
| 1 | 3 | - |
| 2 | 4 | 1 |
| 3 | 6 | 2 |
| 4 | 9 | 3 |
| 5 | 13 | 4 |
| 6 | 18 | 5 |
The first differences increase by exactly 1 each frame. They are not constant, which is the whole point of the item: the next first difference is 6, so frame 7 holds 18 + 6 = 24 squares. A constant-second-difference series is the standard Level G form; the increments themselves form a simple run.
The polygon rule is independent and simpler: sides go 3, 4, 5, 6, 7, 8, so frame 7 is enclosed by a 9-sided regular polygon, a nonagon. Both channels must be right for the option to be right, which is what separates C from D.
2. Explain exactly where Leo's arithmetic went wrong, and construct the check that would have caught it.
Read the worked answer
Leo read the difference column correctly - 1, 2, 3, 4, 5 - and then froze the last value instead of extending the pattern. He treated a growing difference as a constant one at the final step, which is the most common failure on second-difference items: the sequence looks linear for exactly as long as you stop looking.
The check is to write the number under the number before choosing. If the differences are 1, 2, 3, 4, 5, then the rule generating them is "add one more than last time," so the next entry in that column is 6, not 5. Recomputing 18 + 6 takes three seconds and makes the error impossible.
A second check is a plausibility test on the shape of the curve. If the gaps were about to stop growing, the jump from frame 5 to frame 6 would have shrunk or flattened. It did not - it rose from 4 to 5. Nothing in the display signals a brake, so any rule that silently applies one should be suspected.
3. Only frames 1 to 3 were shown. Show that those frames fit a second, quite different rule, name it, and say which frame first rules it out.
Read the worked answer
Frames 1 to 3 hold 3, 4 and 6 squares. Two rules fit all three data points, so the item is genuinely ambiguous at that stage.
Rule one, the intended one: start at 3 and add 1, then 2, then 3, and so on. Next value: 6 + 3 = 9.
Rule two, multiply by 2 and subtract 2: 3 × 2 − 2 = 4, and 4 × 2 − 2 = 6. Both steps check out against the three frames. Next value: 6 × 2 − 2 = 10.
The rules diverge immediately: frame 4 would hold 9 squares under rule one and 10 under rule two. The revealed frame 4 holds 9, so rule two dies there and the solver can proceed with confidence.
The general lesson is that three terms never determine a series. A well-built item shows enough frames that the rival rules have already separated before the blank, and a solver who checks the rivals at frame 4 rather than assuming the first fit is the only fit will not carry a wrong rule into the answer.