Concept 4 of 6
Figure Series: Find the Rule for Any Figure
Today's big idea
A steady gap is what multiplies n; figure 1 then pins what to add.
A steady gap is the multiplier of n; the first figure fixes the number added on.
How it works
👀 What's going on?
At Class 4 a growing pattern asks for figure 5, and adding the gap on four times gets you there. At Class 5 it asks for figure 30. Adding the gap thirty times is not a method, it is a punishment. You want a rule that jumps straight to any figure.
💡 Here's the trick
Look at the matchstick figures above. Their stick counts are 4, 7, 10, 13, rising by 3 every time. That steady gap of 3 is what multiplies the figure number, so the rule looks like 3n plus something. Figure 1 holds 4 and 3 times 1 is 3, so the something is 1 and the rule is 3n + 1. Figure 20 therefore holds 61 sticks, with nothing drawn.
🪄 How to do it
- 1.Count the parts in each figure and write the counts in a row, as the figures above give 4, 7, 10, 13.
- 2.Take the gap between them, because a steady gap is the number that multiplies n.
- 3.Test on figure 1 to pin down what to add, then use the rule on whatever figure is asked.
✍️ Try it
One pattern uses 3, 5, 7, 9 sticks. How many in figure 12? Another holds 1, 4, 9, 16 dots. How many in figure 10?
The first has a gap of 2, so the rule is 2n plus something, and figure 1 holding 3 makes it 2n + 1. Figure 12 uses 25 sticks. The second has gaps of 3, 5 and 7, which never settle, so there is no single multiplier. Those are the square numbers, n times n, and figure 10 holds 100 dots.
⚠️ Watch out
You might be thinking every growing pattern has a steady gap. When the gaps themselves grow, the rule is not n times something. Look at the gaps of the gaps, because 3, 5, 7 rising by 2 each time is the signature of square numbers.
🌟 Why this matters
Seats in a fan-shaped hall, tiles along a border and tins in a stacked display all grow like this. The shop needs the count for row 40, not row 4.
Exam tip: Test your rule on figure 1 before you trust it, because a rule that misses the first figure will miss every figure.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“A pattern uses 6, 11, 16, 21 sticks. How many sticks in figure 20?”
- 1
Find the gap
11 − 6, 16 − 11 and 21 − 16 all give 5. A steady gap means the rule is 5n plus something. 📐
- 2
Pin down what to add, using figure 1
5 × 1 = 5, but figure 1 holds 6, so the rule is 5n + 1. Check it on figure 3: 5 × 3 + 1 = 16 ✓
- 3
Jump straight to figure 20
5 × 20 + 1 = 101, with nothing drawn and nothing counted on.
- 4
Answer ✓
101 sticks
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Find the rule that reaches ANY figure, then use it — no drawing.
RULE FINDER · ROUND 1 OF 3
Step 1 · name the rule
Quick check
✅ VERIFICATION
A pattern uses 8, 13, 18, 23 sticks. How many sticks does figure 12 use?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Shape Series
Split the features, use remainders, then find where both land home.