Repeating Patterns

Look around and patterns are everywhere: the stripes on a jumper, beads threaded on a necklace, red-blue-red-blue lights blinking on a bike, or the tiles on the bathroom floor. Spotting the little bit that keeps coming back lets you say what comes next — and that is a real maths superpower.

A repeating pattern is built from a small group that repeats over and over. That little group is called the core (some people call it the unit). Once you spot the core, you can carry the pattern on forever — and say what comes next without drawing every single step.

In circle, square, circle, square, circle, square, … the core is circle, square. It just keeps repeating, so the next shape must be a circle, then a square, and on it goes.

Patterns don't have to be made of shapes. The thing that repeats can be a colour (red, blue, red, blue, …), a size (big, small, big, small, …), a letter (A, B, B, A, B, B, …), or even a sound (clap, stamp, clap, stamp, …). The maths is exactly the same every time: find the core, then repeat it.

Because it turns a long, scary row into one tiny idea you can hold in your head. You don't have to remember a hundred shapes — you remember the core, and the core builds the whole row for you. That is the big trick behind nearly all of maths: find the small thing that repeats.

Patterns are everywhere

Imagine laying out a snack tray, one fruit at a time: apple, banana, apple, banana, … The core is just apple, banana — two pieces of fruit that keep coming back:

apple banana apple banana apple banana

What comes next? The row ends on a banana, so we are back at the start of the core — the next fruit is an apple.

Cores don't have to be two long. Here is a core that is three long — apple, apple, banana — repeating:

apple apple banana apple apple banana

We call this an A B B pattern: one thing, then a different thing twice. The trick is to keep your eye on where the core starts again — right after every banana.

Find the core, then carry on

Step through the pattern below. First find the core, then repeat it to work out what comes next.

the core is the group of shapes (or colours, sizes, or numbers) that repeats;
to continue a pattern, repeat the core from where it left off;
a growing pattern changes by the same amount each step instead of repeating — like 2, 4, 6, 8, \dots (add 2 each time). That idea grows up into the term-to-term rule of a number sequence.

Three things to do with a pattern

1. Continue it. Here the core is sun, sun, star (three long). To carry on, repeat the core — the next picture is a sun:

sun sun sun sun → ?

2. Fill a gap. One picture is hidden. Read the core (apple, banana) and you know the missing one must be a banana:

apple banana apple [ ? ] apple banana

3. Predict a faraway one. In sun, star, sun, star, … the core is two long. Count in repeats: positions 2, 4, 6, 8, 10 all land on the second shape of the core — a star. So the 10th picture is a star, and you never had to draw all ten.

Two traps that catch people out:

Find where the core restarts. It is easy to glance at \text{A B B A B B} and call the core "A B" — but it restarts every three steps, so the core is three long (A B B), not two.
To predict a faraway item, count in repeats rather than drawing them all. If the core is 3 long, then positions 3, 6, 9, 12, … all land on the last shape of the core.

Spot the core yourself

Here is a row of shapes following a secret repeating core — sometimes two long, sometimes three. The glowing shape on the right is the next one, found by repeating the core. Press Refresh for a brand-new pattern, and try to spot the core before you read the label.