Cycles, not staircases

A pattern that comes back round

The distinction worth making first, because it decides how the question is worked:

A series changes by the same amount and keeps going — bigger, further round, one more each time. A staircase.

A repeating pattern returns to where it started. A cycle.

And that has a very practical consequence. In a repeating pattern, once a whole cycle is on the page, the answer has already appeared earlier in the row. You are not constructing a new figure. You are pointing at one that is already there.

Find the cycle first

Look for the first figure that appears twice — the whole figure, every feature matching — and count how far apart the two copies are. That distance is the length of the cycle. If only one feature has come round again, the shape but not the shading, you have found that feature's cycle and not yet the whole pattern's.

Most are two, three or four long.

Once you have it, the question turns into counting. If the cycle is three and the gap is the seventh box, then the gap holds the same figure as the fourth — and as the first.

Two cycles at once

The harder version runs two cycles of different lengths at the same time — the shape cycling every three, the shading every two.

They come back into step only occasionally, which is exactly what makes the row look irregular when it is nothing of the kind.

Here it matters to separate one feature repeating from the whole figure repeating. A cycle of three and a cycle of two only line up again after six steps, so the complete figure repeats every six. Say the shapes run triangle, square, circle and the fill runs filled, unfilled. The row goes: filled triangle, unfilled square, filled circle, unfilled triangle, filled square — and the sixth is an unfilled circle, which has not appeared anywhere in those five. Its shape has appeared, and its fill has appeared, but not together. The seventh is a filled triangle again, and the whole pattern restarts.

The method does not change: solve one cycle, ignore it completely, solve the other, then combine.

The free check

Before committing: has this figure appeared earlier in the row?

With a single cycle, and a row at least one cycle long, it will have. If it hasn't, one of two things is true — the cycle length is wrong, or this is not a repeating pattern at all and should be treated as a series.

With two cycles at once, ask the question of each feature separately: has this shape appeared, and has this shading appeared? The complete figure may be new, as the unfilled circle was above.

Two seconds, and it catches the commonest error on the type.

Where do pupils lose marks?

Treating a cycle as a staircase, and continuing to grow something that should have reset.

Getting the cycle length wrong, usually by one, which puts every subsequent position out.

Losing count along a long row, which numbering the boxes prevents.

Missing the second cycle when two are running at different lengths.

How can we practise this at home?

Beads, bricks, buttons, anything in a line. Lay out a repeating pattern, cover one, and ask what is underneath — then ask how they know, because "it's the same as that one" is the right answer and the whole idea.

Then make one with two cycles at once — colour repeating every two, shape every three — and watch the moment a child realises the row is not as complicated as it looks.