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Finding the mistake

Patterns and sequences · level 1, lesson 3 of 6

A question to start

Leo made a bead bracelet. He wanted an orange, blue pattern. But something went wrong.

Can you find the bead in the wrong place? How did you know?

Most people find it very fast. That is because your brain expects the pattern to go on. When it does not, the wrong bead jumps out at you, like a wrong note in a song you know well. In this lesson you will learn to find mistakes carefully, every time, even in long and tricky patterns, and to explain how you fixed them.

The idea

A mistake is a place where the rule breaks

A repeating pattern has a rule: say the core again and again. A mistake is a place where the beads do not follow the rule.

In Leo's bracelet, the core is orange, blue. Let us number the beads and say the core along with them: orange (1), blue (2), orange (3), blue (4), orange (5), blue (6)... but bead 6 is orange!

1 2 3 4 5 6 7 8 9 10

So bead 6 is the mistake. To fix it, change it to blue. Then every bead follows the rule.

Explain the fix with "the core is ..."

When you fix a pattern, it is not enough to say "this one is wrong". Say why. A good explanation has two parts:

  1. The core is orange, blue.
  2. So bead 6 should be blue, but it is orange. I changed it to blue.

Here is why the first part matters. Someone else might think the core is something different. If you say what you think the core is, they can check your thinking. Mathematicians always explain their reasons, so other people can check them. That is a big part of what makes mathematics trustworthy.

How to check a long pattern: say the core out loud

A short pattern you can check by looking. A long pattern needs a careful way. Here is one that always works.

  1. Find the core from the start of the pattern. Look at the first few beads and find the part that repeats.
  2. Say the core out loud, again and again, and touch each bead as you say it.
  3. Stop when what you say does not match the bead you touch. That is the mistake.

Here is the method on an AAB pattern. The top row is what the core says should happen. The bottom row is what is really there.

core beads

Go along both rows together. Orange and orange: match. Orange and orange: match. Blue and blue: match. Orange and orange: match. Then the top says orange, but the bottom has blue. That is the mistake: the 5th bead. After that, everything matches again.

Three kinds of mistake

Not every mistake is a wrong colour. There are three kinds.

1. A wrong bead. One bead is the wrong colour or shape, like Leo's bracelet. Fix it by changing that one bead.

2. A missing bead. One bead was left out. Look at this square, triangle, circle pattern:

Square, triangle, circle. Square, triangle, then square again! The circle after the second triangle is missing. Everything after that is still in the right order; it has just slid along by one place. Fix it by putting in an orange circle after the 5th shape.

3. An extra bead. One bead too many was put in.

Orange, blue, orange, blue, then blue again. The 5th bead is extra. Take it out, and the rest goes orange, blue, orange, blue, as it should.

Why "the smallest fix" is the best fix

Look again at the bracelet with the extra blue bead. Someone might try to fix it by changing colours instead of taking a bead out. They would have to change bead 5 to orange, bead 6 to blue, bead 7 to orange, bead 8 to blue and bead 9 to orange. That is 5 changes! Taking out the one extra bead is 1 change.

Both ways end with a correct pattern. But one change is much more likely to be what really went wrong. When someone makes a bracelet, it is more likely they slipped once than that they made five mistakes in a row. So we look for the smallest fix: the fewest changes that make the whole pattern follow the rule.

This is a real idea used by grown-ups. When a computer sends a message and a bit of it gets spoiled on the way, the computer at the other end looks for the smallest change that makes the message make sense again.

Two beads swapped

Sometimes two beads next to each other have swapped places. In a square, triangle, circle pattern you might see square, circle, triangle. Two beads are wrong, but it was one slip: the two were put on in the wrong order. Fix it by swapping them back.

Can a pattern have no mistake at all, but still look odd?

Yes! Sometimes a pattern looks wrong but is fine, because the core is longer than you thought. Look:

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

At first it looks like orange, orange, blue with a mistake. But read on: orange, orange, blue, orange, blue comes twice. The core is 5 beads long: orange, orange, blue, orange, blue. No mistake at all!

So before you call something a mistake, look at all the beads, not only the first few. A mistake happens once. A core comes again and again.

Worked examples

Example 1. One wrong bead. Find and fix the mistake.

  1. From the start: green, orange, green, orange. The core is green, orange.
  2. Say it along the beads: green (1) ✓, orange (2) ✓, green (3) ✓, orange (4) ✓, green (5)... but bead 5 is orange. ✗
  3. Keep going to check the rest: orange (6) ✓, green (7) ✓, orange (8) ✓.
  4. The core is green, orange, so bead 5 should be green. Change it to green.

Example 2. Is it a mistake? A row of stickers goes cat, dog, dog, cat, dog, dog, cat, dog, dog. Ana says the 3rd sticker is a mistake because "it should be cat, dog, cat, dog". Is she right?

  1. Look at all the stickers. Cat, dog, dog comes three times.
  2. So the core is cat, dog, dog, an ABB pattern. Every sticker follows it.
  3. Ana is not right. She only looked at the first two stickers and guessed the core too soon. There is no mistake.

Example 3. A missing bead. Find and fix the mistake.

  1. The core from the start: orange triangle, orange triangle, blue square. (AAB.)
  2. Check: shapes 1, 2, 3 ✓. Shape 4 is an orange triangle ✓. Shape 5 should be an orange triangle, but it is a blue square ✗.
  3. Should we change shape 5? Look further. After it come: triangle, triangle, square, triangle, triangle, square. That is perfect AAB. If we changed shape 5 to a triangle, we would get triangle, triangle, triangle in a row, and then more would be wrong.
  4. So one triangle is missing after shape 4. Put in one orange triangle there, and the whole row is triangle, triangle, square four times.
  5. The core is triangle, triangle, square, so after the 4th shape another triangle is needed. One change: add it.

Try it

Questions 1 to 3 use these rows.

Row 1.

Row 2.

Row 3.

  1. Find the mistake in row 1. Explain your fix with "the core is ...".
  2. Find the mistake in row 2. Explain your fix.
  3. Something went wrong in row 3. What kind of mistake is it, and how do you fix it?
  4. Find the mistake: Monday, Tuesday, Wednesday, Friday, Saturday, Sunday.
  5. Find the mistake: clap, stamp, clap, stamp, clap, clap, stamp, clap, stamp.
  6. Is there a mistake? red, red, blue, red, red, blue, red, red, blue, red, red.
  7. A row goes A B C A B C A B A B C. What is wrong, and what is the smallest fix?
  8. Make a pattern with one mistake in it, and ask someone at home to find it. Can they explain the fix?
Answers
  1. Bead 5. The core is blue, green, so bead 5 should be blue, but it is green. Change it to blue.
  2. Shape 8. The core is square, square, triangle (orange, orange, blue). So shapes 7 and 8 should both be orange squares, and shape 9 a blue triangle. Shape 8 is a blue triangle, so change it to an orange square.
  3. Two shapes have swapped places. The core is circle, square, triangle. Shapes 7, 8, 9 are circle, triangle, square, so the 8th and 9th are the wrong way round. Swap them back: circle, square, triangle.
  4. Thursday is missing. The days of the week go Monday, Tuesday, Wednesday, Thursday, Friday... Put Thursday in after Wednesday. (This is a missing-bead mistake.)
  5. There is one clap too many. The core is clap, stamp. Sounds 1 to 5 are clap, stamp, clap, stamp, clap. The 6th should be stamp, but it is clap. The rest after it (stamp, clap, stamp) fits if we take out that extra clap. Smallest fix: take out the 6th sound.
  6. No mistake. The core is red, red, blue, and the row just stops in the middle of a core. The next bead would be blue.
  7. After A B C A B C A B the next should be C, but it is A. Then A B C follows correctly. So a C is missing after the 8th letter. Smallest fix: put in one C. (Changing letters instead would take more changes.)
  8. Any pattern works. A good one has a core that repeats at least three times, so the mistake is easy to prove.

Common mistake

Calling the first odd-looking bead a mistake, without looking at the rest. The core may be longer than you think, or the pattern may just end in the middle of a core. Always check the whole row before you decide.

Changing many beads when one change would do. If a whole lot of beads after one spot look wrong, the real mistake is probably a missing or extra bead at that spot. Look for the smallest fix.

Where it leads