Four number lists, each built by its own rule: (1) 2, □, 10, 14, 18, 22, … (2) 3, 6, 9, 2, 3, □, 9, 2, 3, 6, … (3) 2, 3, 5, 8, 12, □, □, … (4) 1, 2, 5, 14, 41, □, 365, …. Work out each rule and fill in the boxes.
Your answer
Watch outReusing the first rule is tempting, but all four differ — read each list afresh.
How to solve
StrategyLook for a Pattern — The rule is hidden inside the numbers, so the first move on every list is the same: write the gap between each pair of neighbours in a row underneath. A row of equal gaps means adding the same amount each time; a row of gaps that itself grows steadily means the step is growing; a row of gaps that goes nowhere is a hint that the list repeats rather than grows, and then I count how long the repeating block is instead. Whatever rule I settle on I check against every printed number before writing anything in the box, and in list (4) the number printed after the box makes that check completely certain.
1STEP 1
Set up the same first move for every list
For each list, write down the gaps between neighbours.
Looking at how much the list jumps each time, rather than at the numbers themselves, is what makes an invisible rule visible.
4.OA.C.5Look For A Pattern
2STEP 2
List (1): find the constant step
List (1) adds 4 each time, so the box is 6.
2 + 4 = 6, 6 + 4 = 10
With a constant step you can walk forwards from the number before the box and land on the number after it, so the box is confirmed from both sides at once.
4.OA.C.5Look For A Pattern
With a constant step you can walk forward from the number before the box and land on the number after it, confirming the box from both sides.
Why?
Every term is the one before it plus the same fixed jump, so any term can be reached from a neighbour by one jump.
🧱A constant step reaches any termWhen a list climbs by the same amount every time, any term is the starting value plus that step repeated, so you never have to write out the ones between.
Why?
Stepping back from the number after the box undoes stepping forward, so the two routes must land on the same value.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
3STEP 3
List (2): notice it repeats instead of growing
List (2) does not grow — four numbers repeat.
3, 6, 9, 2 ∣ 3, 6, 9, 2 ∣ 3, 6
Writing the list in rows of four lines the repeats up in columns, which shows the block length far more reliably than guessing where the pattern restarts.
4.OA.C.5Make A Systematic List
4STEP 4
List (2): read off the box
Counting round the cycle, (2)'s box is 6.
6 = 4 + 2
Once you know the block length, finding any position means stepping back by whole blocks until you land inside the first one — plain counting, no formula needed.
4.OA.C.5Make A Systematic List
5STEP 5
List (3): find the growing step
In (3) the gap grows by 1, so the boxes are 17 and 23.
12 + 5 = 17, 17 + 6 = 23
When the numbers themselves have no obvious rule, the gaps often do, and here the gaps form the counting numbers a Grade 4 student spots instantly.
4.OA.C.5Look For A Pattern
6STEP 6
List (4): look at the gaps first
In (4) the gap triples, so the box is 122.
1, 3, 9, 27, 81 41 + 81 = 122
Tripling gaps is easy to see once the gaps are written out, and it needs nothing harder than multiplying by 3.
4.NBT.B.5Look For A Pattern
7STEP 7
List (4): check against the printed 365
Reaching the printed 365 confirms the rule is right.
A number printed on the far side of the box is a free test: a guessed rule that also lands on 365 is almost certainly the intended one.
4.NBT.B.5Guess And Check
Answer
6, 6, 17 and 23, 122
Each answer sits sensibly between its neighbours: 6 is between 2 and 10 in list (1); 6 is a number that already appears in the repeating block in list (2); 17 and 23 continue a list whose steps are getting bigger, so they are bigger jumps than the 4 before them; and 122 sits between 41 and 365, which is right for a list whose steps keep tripling. Every rule was also tested on numbers other than the ones beside the box — list (1) on three separate gaps, list (2) on the printed 7th, 8th, 9th and 10th numbers, list (3) on all four printed gaps, and list (4) on the printed 365 — so none of them was fitted to the box alone.
Takeaway
Write the gaps under the numbers first — equal gaps mean add the same amount, growing gaps mean the step is growing, and no useful gaps at all means the list is just repeating a short block.