Skip to content

Making your own pattern

Patterns and sequences · level 1, lesson 4 of 6

A question to start

Lily made a pattern for her friend Omar to guess.

?

"What comes next?" she asked. Omar said "orange". Lily said "No! It is green!" Omar said "That's not fair!"

Who is right? Can Omar be sure what comes next?

Think about it. Omar thought the core was green, orange, orange. Then the 4th bead (green) starts the core again, and the next bead is orange. That is a sensible guess! But Lily's core was really green, orange, orange, green, 4 beads long, so the next bead starts a new core: green. Both rules fit the four beads Omar could see. He could not know which one Lily meant, because she showed her core only once. In this lesson you will learn how to make a pattern that is fun, and also fair.

The idea

Three steps to make a pattern

Making a repeating pattern takes three steps.

  1. Choose your things. Beads, blocks, stickers, claps, colours, shapes, letters, toys. Two or three kinds is plenty.
  2. Choose your core. Decide what order the things go in, and how long the core is.
  3. Repeat the core at least three times. Say it, build it, check it.

Here the things are blue squares and green triangles. The core is square, square, triangle. It is repeated three times, and each core is marked with a bracket.

Why three times?

If you show your core only once, nobody can tell it is a core. It could be the start of anything. That is what went wrong for Lily.

If you show it twice, your friend can guess the rule. If you show it three times, your friend can check the guess: "It went square, square, triangle, and again, and again. I am sure now."

This is a good rule for all of mathematics, not only for beads. When you think you see a pattern, look at more examples before you trust it. Seeing something once is a guess. Seeing it come back again and again is evidence.

How many cores can you make?

Suppose you have orange beads and blue beads, and you want a core 3 beads long that uses both colours. How many different cores are there?

Let us be organised. The first bead is orange or blue. The second is orange or blue. The third is orange or blue. Here are all the ones that use both colours:

AAB ABA ABB ABB ABA AAB

There are 6. (Orange, orange, orange and blue, blue, blue use only one colour, so we left them out.) The letter names beside them show something interesting: each letter name comes twice. Orange, orange, blue and blue, blue, orange are both AAB. They are the same pattern with the colours swapped.

So with two colours and a core of 3, there are only 3 different pattern shapes: AAB, ABA, ABB.

A surprise: the same row, starting in a different place

Look at the ABA core: orange, blue, orange. Repeat it:

orange, blue, orange, orange, blue, orange, orange, blue, orange

Now look closely at the middle of that row. Do you see orange, orange, blue, orange, orange, blue? It is the AAB pattern hiding inside!

core AAB core ABA

The bottom row is the ABA pattern, moved along one place. From the second bead on, it is exactly the same as the top row, the AAB pattern. A long string of repeating beads does not really have a "start". If you made it into a bracelet (a circle), you could not even tell where the core begins. The rule is the same; only the starting place changes.

That is why mathematicians care most about the core and its length, more than where you start.

Making a pattern fair, and making it tricky

A good puzzle is tricky but fair. Here is how.

To make it fair:

  • Show the core at least three times.
  • Use beads that are easy to tell apart.
  • Make sure every core is exactly the same. Check it before you show anyone!

To make it trickier (but still fair):

  • Use a longer core. Red, red, blue, green (4 long) is harder than red, blue (2 long).
  • Use the same thing more than once in the core, in different places: red, blue, red, red (ABAA).
  • Stop in the middle of a core, so the next bead is not the first one.
  • Use two kinds of change at once: a pattern of colours and sizes, like big red, small red, big blue, small blue.

Patterns that go across and down

You can also make a pattern on a grid, like a quilt or floor tiles. Here are two.

On the left, each row is an AB pattern (orange, blue), and each row starts with the other colour. That makes a chessboard. Read down any column and you get an AB pattern too!

On the right, each row is an ABC pattern: orange, blue, green. Every row starts the same way, so you get stripes going down.

When you make a grid pattern, there are two rules: one for going across, and one for how each row starts.

Worked examples

Example 1. Make a fair AAB pattern with stickers. Use stars and hearts.

  1. Things: stars and hearts.
  2. Core: AAB, so star, star, heart.
  3. Repeat three times: star, star, heart, star, star, heart, star, star, heart.
  4. Check each core: star, star, heart ✓ ✓ ✓.

Your friend sees star, star, heart three times and can be sure that a star comes next.

Example 2. Fix an unfair pattern. Tom showed: red, red, blue, green, red. "What comes next?" Is it fair?

  1. Tom has shown red, red, blue, green, then red. Maybe the core is red, red, blue, green, and the next bead is red. But we have only seen that core once, plus one bead.
  2. It is not fair yet. To make it fair, Tom should show red, red, blue, green, red, red, blue, green, red, red, blue, green: the core three times. Then the next bead is surely red.

Example 3. All the cores 2 long. You have a red block and a yellow block (and lots more of each). Which cores of length 2 use both colours?

  1. First block red or yellow; second block red or yellow.
  2. Red, yellow ✓. Yellow, red ✓. (Red, red and yellow, yellow use only one colour.)
  3. So there are 2 cores: red, yellow and yellow, red. Both are AB. Repeat either one, and you get the same long row, just starting in a different place.

Example 4. A two-change pattern. Make a pattern that changes colour and size.

  1. Things: big and small circles, orange and blue.
  2. Core: big orange, small orange, big blue, small blue.
  3. Repeat: big orange, small orange, big blue, small blue, big orange, small orange, big blue, small blue, big orange, small orange, big blue, small blue.
  4. Look at the sizes alone: big, small, big, small... an AB pattern. Look at the colours alone: orange, orange, blue, blue... an AABB pattern. Two patterns are going on at the same time!

Try it

  1. Make an ABC pattern with three kinds of fruit. Show the core three times.
  2. Make an ABB pattern with claps and stamps. Say it out loud.
  3. Sara showed: cat, dog, cat. "What comes next?" Explain why her friend cannot be sure.
  4. Lists of beads 3 long with two colours: we found 6 that use both colours. Using green and orange, write all 6.
  5. Make a pattern that is 4 long and uses three colours.
  6. Look at the chessboard quilt. What colour is the square in the 4th row and the 6th column? (Count rows from the top and columns from the left.)
  7. Make a pattern that stops in the middle of a core. Ask: what comes next?
  8. Here is a bracelet (a circle) of beads: orange, blue, blue, orange, blue, blue, and then back to the start. Mo says the core is orange, blue, blue. Jen says the core is blue, blue, orange. Who is right?
Answers
  1. For example apple, pear, banana, apple, pear, banana, apple, pear, banana. Any three different fruits work.
  2. For example clap, stamp, stamp, clap, stamp, stamp, clap, stamp, stamp.
  3. The core was shown only once (or not even once). The next could be dog (cat, dog), or cat (cat, dog, cat, cat), or something else. Sara should show the core three times.
  4. green green orange; green orange green; green orange orange; orange green green; orange green orange; orange orange green.
  5. For example red, blue, green, blue, red, blue, green, blue... The core red, blue, green, blue is 4 long and uses three colours (ABCB).
  6. In the chessboard, row 1 starts orange, row 2 starts blue, row 3 orange, row 4 blue. Row 4 goes blue, orange, blue, orange, blue, orange. So the square in row 4, column 6 is orange. (Column 6 is even and row 4 starts blue, so the even places in row 4 are orange.)
  7. For example: red, red, blue, red, red, blue, red, red, blue, red. Next is red, because the last core has only started.
  8. Both! On a bracelet there is no first bead. Going round, you see orange, blue, blue, orange, blue, blue. If you start at a blue bead, you see blue, blue, orange, blue, blue, orange. It is the same circle of beads. Both cores are 3 long, and both describe it.

Common mistake

Showing the core only once. Then it is a guess, not a pattern your friend can work out. Repeat it at least three times.

Making a mistake in your own pattern. When you build a long pattern, it is easy to slip. Say the core out loud as you build, and check every core at the end (like in Finding the mistake).

Where it leads