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Download complete ZIPPuzzles · difficulty 1–5
Number sequences
Twenty-five original runs of numbers, each with one term blanked out, from a single repeated step up to two rules interleaved. The method is the point: difference tables, ratio checks, splitting a run by position, and the discipline to test a rule against every printed term before trusting it.
What this category trains
Inducing a rule from a short run
First, second and third difference analysis
Recursive and interleaved rules
Resisting the first plausible rule
25 puzzles · about 50 minutes end to end, as a guide only · untimed
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The ladder
The puzzles below run in difficulty order, easiest first. Each one checks on its own and shows its explanation straight away, so you get the method for one rung before you attempt the next.
Work the ladder from the top: the early runs fix one habit each, and the later ones combine those habits. Nothing here is timed and nothing is scored. Write the differences between neighbouring terms underneath the run before you guess, that single move solves a surprising share of them, and read the explanation even when you were right, because it names the method you can reach for next time.
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1. Steady steps upward
Each term sits the same distance above the one before it. Which number belongs where the question mark is?
Six terms in order, the last one missing.
- 4
- 7
- 10
- 13
- 16
- missing term, position 6 of 6
Show a hint
Write the gap between each pair of neighbouring terms underneath the run and check they all match.
2. Sevens going down
This run falls by the same amount at every step. Which number takes the place of the question mark?
A descending run of five terms.
- 48
- 41
- 34
- 27
- missing term, position 5 of 5
Show a hint
Subtract each term from the one before it, and count the size of the gap rather than the numbers inside it.
3. Twice as big each time
The gaps in this run grow, so subtraction will not settle it. Try comparing each term with the one before it by division. What replaces the question mark?
Five terms in order, the last one missing.
- 3
- 6
- 12
- 24
- missing term, position 5 of 5
Show a hint
Divide each term by the one before it; if those results agree, you have the rule.
4. The gap in the middle
The blank sits inside the run rather than at the end, and every step is the same size. Which number is missing?
Six terms in order, with the fourth missing.
- 5
- 11
- 17
- missing term, position 4 of 6
- 29
- 35
Show a hint
Find the step from a pair you can see, apply it once to the term on the left of the blank, then check forwards.
5. Straight through zero
This run keeps falling by the same amount, and it does not pause when it passes zero. Which number belongs in the blank?
Six terms in order, with the fourth missing.
- 14
- 9
- 4
- missing term, position 4 of 6
- −6
- −11
Show a hint
Carry on subtracting the same amount past zero instead of treating zero as a floor the run must land on.
6. Three times over
Each term is a fixed multiple of the one before it. Which value replaces the question mark at the end of the run?
Five terms in order, the last one missing.
- 2
- 6
- 18
- 54
- missing term, position 5 of 5
Show a hint
Work out the multiplier from two different pairs of neighbours before you use it on the blank.
7. Two runs in one line
The terms here jump up and down, because this line is really two separate runs written alternately. Which number belongs where the question mark is?
Eight terms in order, with the seventh missing.
- 3
- 20
- 7
- 18
- 11
- 16
- missing term, position 7 of 8
- 14
Show a hint
Read every other term as a run of its own, starting from the very first term.
8. Gaps that grow steadily
The steps in this run are not equal, but they change in a regular way. Which number completes it?
Six terms in order, the last one missing.
- 3
- 5
- 9
- 15
- 23
- missing term, position 6 of 6
Show a hint
Write the differences in one row, then take the differences of that row as well.
9. Squares with one missing
Every term in this run is a whole number multiplied by itself. Which value is missing from the middle?
Six terms in order, with the fourth missing.
- 16
- 25
- 36
- missing term, position 4 of 6
- 64
- 81
Show a hint
Ask what each term is the square of, and write those roots in a row of their own.
10. Each term leans on two
No single step size or multiplier fits this run. Look instead at what any two neighbouring terms make together. What is the missing value?
Six terms in order, the last one missing.
- 4
- 7
- 11
- 18
- 29
- missing term, position 6 of 6
Show a hint
Add each pair of neighbouring terms and see whether the total is already sitting in the run.
11. Double, then one more
Each term comes from the one before it by the same two-part instruction. Which number finishes the run?
Five terms in order, the last one missing.
- 3
- 7
- 15
- 31
- missing term, position 5 of 5
Show a hint
State the rule as a sentence with the two operations in order, then apply it exactly once.
12. Two rows of differences
One row of differences will not settle this run. Which number belongs at the end of it?
Six terms in order, the last one missing.
- 1
- 3
- 8
- 17
- 31
- missing term, position 6 of 6
Show a hint
Difference the run, then difference that row too, and see how the second row is changing.
13. Add five, then double
Two operations take turns in this run. Which number comes next?
Seven terms in order, the last one missing.
- 3
- 8
- 16
- 21
- 42
- 47
- missing term, position 7 of 7
Show a hint
Label each step with the operation that produced it, then read off which operation is due.
14. Cubes with a little extra
These terms grow faster than squares do. Which value completes the run?
Six terms in order, the last one missing.
- 3
- 10
- 29
- 66
- 127
- missing term, position 6 of 6
Show a hint
Write the cubes 1, 8, 27, 64, 125 beneath the run and look at the difference between the rows.
15. Position times its neighbour
Compare each term with the place it occupies in the run rather than with the term before it. Which number belongs at the end?
Six terms in order, the last one missing.
- 2
- 6
- 12
- 20
- 30
- missing term, position 6 of 6
Show a hint
Number the places 1, 2, 3 and ask what each term is made of once you know its place.
16. One run doubles, one climbs
Two rules share this line: one governs the terms in the odd places and one the terms in the even places. Which value is missing?
Eight terms in order, with the sixth missing.
- 2
- 5
- 4
- 14
- 8
- missing term, position 6 of 8
- 16
- 32
Show a hint
The terms in the even places form a run of their own, and the last of them is already printed.
17. One bigger every time
The steps in this run grow by one at each stage. Which number sits in the blank?
Seven terms in order, with the fifth missing.
- 1
- 3
- 6
- 10
- missing term, position 5 of 7
- 21
- 28
Show a hint
Work out the gaps you can see, continue the pattern in the gaps, then check your value against the terms on the right.
18. A multiplier that will not sit still
The terms here are multiplied at every step, but not by the same amount each time. Which value comes next?
Six terms in order, the last one missing.
- 2
- 4
- 12
- 48
- 240
- missing term, position 6 of 6
Show a hint
Divide each term by the one before it, then treat those results as a little run in their own right.
19. The term feeds itself
Nothing constant is added here and no term is a neat multiple of the one before it. Look at the digits of each term. Which number completes the run?
Six terms in order, the last one missing.
- 48
- 60
- 66
- 78
- 93
- missing term, position 6 of 6
Show a hint
Add the digits of each term together and compare that total with the step that follows it.
20. A familiar list in disguise
Subtracting the same constant from every term uncovers a well-known list of whole numbers. Which value comes next?
Seven terms in order, the last one missing.
- 6
- 7
- 9
- 11
- 15
- 17
- missing term, position 7 of 7
Show a hint
Try subtracting a small constant from every term and see whether the results are a list you already know.
21. Odd places climb, even places leap
The terms in the odd places follow one rule and the terms in the even places follow another. Which number is missing?
Eight terms in order, with the sixth missing.
- 5
- 3
- 11
- 9
- 17
- missing term, position 6 of 8
- 23
- 81
Show a hint
Take the terms in the even places on their own, and use the very last term of the line to test your rule.
22. Looking two terms back
Each term is built from the two before it, but not by simply adding them together. What is the next value?
Seven terms in order, the last one missing.
- 1
- 3
- 5
- 11
- 21
- 43
- missing term, position 7 of 7
Show a hint
Try previous term plus a multiple of the term before that, and test the multiple on every printed term.
23. Three rows down
Two rows of differences are not enough for this run, and the missing term sits inside it. Which value belongs there?
Six terms in order, with the fifth missing.
- 2
- 10
- 30
- 68
- missing term, position 5 of 6
- 222
Show a hint
Keep differencing until a row stops changing, then rebuild the table from the bottom row upwards.
24. Digits and places together
Each step here depends both on the digits of the term it starts from and on how far along the run that step falls. Which value completes it?
Six terms in order, the last one missing.
- 7
- 21
- 30
- 42
- 72
- missing term, position 6 of 6
Show a hint
Divide each step by the digit sum of the term it starts from, and look at what those four results spell out.
25. Two runs of different character
Two runs are interleaved here. One grows out of its own earlier terms; the other depends on the place it occupies. Which number fills the gap?
Ten terms in order, with the sixth missing.
- 3
- 6
- 4
- 9
- 7
- missing term, position 6 of 10
- 11
- 21
- 18
- 30
Show a hint
Number the even places 1, 2, 3, 4, 5 and compare each of those terms with its own place number.
Skills these puzzles practise
Each category maps onto the shared aptitude constructs, so a puzzle that interests you has a lesson and a larger practice bank behind it:
Arithmetic, fractions, percentages, ratios, rates, estimation, word problems, and number relationships.
Identifying general rules from examples, sequences, and observations.
Inferring rules from shapes, symbols, matrices, and non-verbal patterns.
Comparing strings, records, transactions, forms, and data for mistakes.
Holding, updating, and manipulating short sequences, rules, or instructions.
Selecting methods, combining information, troubleshooting, and reaching practical solutions.
Sustained focus, selective attention, and accuracy under time pressure.