Puzzles · difficulty 1–5
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.
25 puzzles · about 50 minutes end to end, as a guide only · untimed
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.
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.
Write the gap between each pair of neighbouring terms underneath the run and check they all match.
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.
Subtract each term from the one before it, and count the size of the gap rather than the numbers inside it.
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.
Divide each term by the one before it; if those results agree, you have the rule.
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.
Find the step from a pair you can see, apply it once to the term on the left of the blank, then check forwards.
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.
Carry on subtracting the same amount past zero instead of treating zero as a floor the run must land on.
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.
Work out the multiplier from two different pairs of neighbours before you use it on the blank.
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.
Read every other term as a run of its own, starting from the very first term.
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.
Write the differences in one row, then take the differences of that row as well.
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.
Ask what each term is the square of, and write those roots in a row of their own.
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.
Add each pair of neighbouring terms and see whether the total is already sitting in the run.
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.
State the rule as a sentence with the two operations in order, then apply it exactly once.
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.
Difference the run, then difference that row too, and see how the second row is changing.
Two operations take turns in this run. Which number comes next?
Seven terms in order, the last one missing.
Label each step with the operation that produced it, then read off which operation is due.
These terms grow faster than squares do. Which value completes the run?
Six terms in order, the last one missing.
Write the cubes 1, 8, 27, 64, 125 beneath the run and look at the difference between the rows.
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.
Number the places 1, 2, 3 and ask what each term is made of once you know its place.
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.
The terms in the even places form a run of their own, and the last of them is already printed.
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.
Work out the gaps you can see, continue the pattern in the gaps, then check your value against the terms on the right.
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.
Divide each term by the one before it, then treat those results as a little run in their own right.
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.
Add the digits of each term together and compare that total with the step that follows it.
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.
Try subtracting a small constant from every term and see whether the results are a list you already know.
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.
Take the terms in the even places on their own, and use the very last term of the line to test your rule.
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.
Try previous term plus a multiple of the term before that, and test the multiple on every printed term.
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.
Keep differencing until a row stops changing, then rebuild the table from the bottom row upwards.
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.
Divide each step by the digit sum of the term it starts from, and look at what those four results spell out.
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.
Number the even places 1, 2, 3, 4, 5 and compare each of those terms with its own place number.
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.
