Reading a sequence: how progressions are built
The trick is never to look at the sequence. It is to look at one attribute of it.
A sequence puzzle shows several figures in order and asks what comes next. The figures usually differ in more than one way at once, which is what makes them hard — not the individual rules, which are almost always simple.
The method is to stop looking at the sequence as a whole and look at exactly one property of it at a time.
The attributes worth checking
Almost every sequence you will meet changes one or more of these:
- Count — how many elements there are
- Size — how large they are
- Position — where in the frame they sit
- Orientation — which way they point
- Shading — filled, empty, striped
- Shape — the form itself, which may cycle
- Number of sides or corners — a numeric rule wearing a picture as a disguise
Go through that list one item at a time. It is slower than staring for the first few puzzles and much faster after that, because you stop missing things.
Count goes 1, 2, 3, so next is 4. Size goes small, medium, large — either it resets or it keeps growing, and you need another frame or the options to decide. Shading went empty, empty, filled, which is the interesting one: two frames of nothing then a change is a rule with a period of three, so frame 4 is probably empty again. Each rule is trivial. Seen all together they look like a mess.
Watch for the step changing
The most common upgrade from an easy sequence to a hard one is not a stranger rule — it is a step that is itself changing.
- Constant step: 2, 4, 6, 8 — add two each time
- Growing step: 2, 4, 8, 16 — double each time, or 1, 2, 4, 7, 11 where the step goes 1, 2, 3, 4
- Alternating step: 1, 4, 2, 5, 3 — two sequences interleaved
If the differences between consecutive terms look irregular, write the differences down and look at those as a sequence. That one move solves a large fraction of otherwise impossible-looking number and count puzzles.
Cycles are not progressions
A progression keeps going. A cycle returns to the start. They look identical for the first few terms, which is precisely the trap.
If shape goes circle, triangle, square and there are four frames, the fourth is probably a circle again rather than a pentagon. Whether a rule cycles or continues is often decided by the answer options rather than the sequence itself — if none of the options is a pentagon but one is a circle, the rule cycles.
Using the options to decide between two readings is not cheating either. It is using all the information the puzzle gave you. What loses marks is looking at the options before you have a reading at all.
Position deserves its own pass
Movement rules are the ones people miss most often, because we look at what a shape is before we look at where it is.
Check whether elements move a fixed number of places each frame, whether they travel around the edge of the frame, whether they wrap around when they reach the end, and whether two elements are moving in opposite directions. Rotation around a frame is extremely common and reads as chaos until you notice it, at which point it is obvious.
A working order
- Count the elements in each frame. Write the numbers down.
- Check position — is anything moving, and by how much?
- Check orientation — is anything turning, and by how much?
- Check shading and colour as its own pass.
- Check shape, and ask whether it progresses or cycles.
- Combine: state each rule in a sentence, then build the next frame from those sentences.
Six passes sounds laborious. In practice it takes under a minute once it is habit, and it is far quicker than the alternative, which is looking at the whole thing for three minutes and then guessing.