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🏅 Olympiad maths · session 1 of 49 · 60 minutes

Number patterns: find the rule

Olympiad questions love patterns. Learn to spot the rule, jump straight to the 50th or 100th term without listing them all, and see why triangular and square numbers grow the way they do.

🖨 Practice sheet

A complete lesson · 60 minutes

Teach it, see it, practise it, all on this page.

Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.

See the teacher's plan
  1. Warm-up4 min
  2. Differences8 min
  3. Shapes9 min
  4. Doubling and Fibonacci7 min
  5. Working backwards7 min
  6. Activity 17 min
  7. Activity 27 min
  8. Activity 36 min
  9. Quiz4 min
  10. Set the project1 min

By the end you can

  • Find the rule of a pattern by looking at differences
  • Find any term (the 20th, the 100th) without writing them all
  • Recognise triangular, square, doubling and Fibonacci patterns
  • Work backwards to find which term a number is

Before you start

  • Times tables
  • Adding and multiplying two-digit numbers

Have ready

  • Paper and pencil
  • About 20 small objects (coins, buttons) for building shapes

Classes 4 to 7 · ages 9 to 13

Warm-up · think first

A lily pad doubles in size every day. On day 20 it covers the whole pond. On which day did it cover half the pond?

Show the answer

Day 19. It doubles every day, so the day before it was full it was half. Thinking backwards is often faster than thinking forwards, and that is a big Olympiad habit.

Teacher tip: Most people say day 10 first. Let the learner say an answer, then ask: what happens between day 19 and day 20?

Part 1 of 4 · learn, see, practise

Look at the differences

To find the rule of a list of numbers, write the gap between each pair of neighbours. If the gap is always the same, the pattern adds that number each time.

To find a term far away, count the jumps. The 20th term is 19 jumps after the 1st term, so it is first term + 19 × gap.

Example 1

What are the next two numbers in 5, 8, 11, 14, …?

  1. The gaps are 3, 3, 3: the rule is add 3.
  2. 14 + 3 = 17, then 17 + 3 = 20.

Answer: 17 and 20

Example 2

What is the 20th number in 5, 8, 11, 14, …?

  1. From the 1st number to the 20th there are 19 jumps.
  2. Each jump adds 3, so we add 19 × 3 = 57.
  3. 5 + 57 = 62.

Answer: 62

Think first: Find the 50th number in 2, 9, 16, 23, … using jumps (do not list them!).

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 24
  2. 34
  3. 123
  4. -17
  5. 4
  6. 50

Part 2 of 4 · learn, see, practise

Growing shapes: triangle and square numbers

Some patterns come from shapes. Stack dots in rows of 1, 2, 3, … and you get triangle numbers: 1, 3, 6, 10, 15, …. Each new row adds one more dot than the last.

Square numbers come from squares of dots: 1, 4, 9, 16, 25, …. The 10th one is 10 × 10.

A clever trick for triangle numbers: two copies of a triangle make a rectangle n by n + 1, so the nth triangle number is n × (n + 1) ÷ 2.

Example 1

How many dots are in the 10th triangle?

  1. It is 1 + 2 + 3 + … + 10.
  2. Pair the numbers from the outside: 1 + 10, 2 + 9, 3 + 8, 4 + 7, 5 + 6. That is five pairs of 11.
  3. 5 × 11 = 55. The rule n × (n + 1) ÷ 2 gives 10 × 11 ÷ 2 = 55 too.

Answer: 55

Build the shapes

Choose a shape and change the figure number. Count the dots, then check the rule underneath.

  1. Make a triangle with exactly 15 dots.
  2. Make a square with 64 dots.
  3. Make a rectangle shape (n by n + 1) with 42 dots.

Think first: Without building it, how many dots are in the 20th triangle? Then check with the visual's rule line.

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 21
  2. 144
  3. 5050
  4. 9
  5. 100
  6. 240

Part 3 of 4 · learn, see, practise

Doubling patterns and Fibonacci

Not every pattern adds. If the gaps keep growing fast, look at what you multiply by: 3, 6, 12, 24 doubles each time.

In the Fibonacci pattern 1, 1, 2, 3, 5, 8, … each number is the sum of the two before it. It appears in sunflowers, pine cones and many Olympiad questions.

Example 1

What comes next: 3, 6, 12, 24, …?

  1. The gaps are 3, 6, 12: they double, so this is not an adding pattern.
  2. Each number is double the one before: 24 × 2 = 48.

Answer: 48

Example 2

Find the next two: 1, 1, 2, 3, 5, 8, …

  1. Each number is the sum of the two before it.
  2. 5 + 8 = 13 and 8 + 13 = 21.

Answer: 13 and 21

Think first: Make up your own doubling pattern that starts at 5 and write its first six numbers.

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 80
  2. 26
  3. 729
  4. 55
  5. 22
  6. 33

Part 4 of 4 · learn, see, practise

Working backwards: which term is it?

Olympiad questions often turn the problem around: they give a number and ask where it is in the pattern.

Take away the first term to see how much was added, divide by the gap to count the jumps, and add 1 because the first term had no jump.

Example 1

Which term of 4, 7, 10, 13, … is 100?

  1. 100 − 4 = 96 was added in total.
  2. Each jump adds 3, so there were 96 ÷ 3 = 32 jumps.
  3. 32 jumps after the 1st term is the 33rd term.

Answer: The 33rd term

Example 2

Squares made of matchsticks in a row need 4, 7, 10, … sticks. How many sticks for 12 squares?

  1. The first square needs 4 sticks; each new square shares a side, so it needs 3 more.
  2. 12 squares is the first square plus 11 more: 4 + 11 × 3 = 37.

Answer: 37 matchsticks

Think first: Which term of 6, 11, 16, 21, … is 101?

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 10
  2. 20
  3. 41
  4. no
  5. 78

Watch out

Common mistakes

Counting terms instead of jumps

From the 1st to the 20th term there are 19 jumps, not 20.

Assuming every pattern adds

Check whether the gaps stay the same. If they grow, try doubling, squares or adding the last two.

Forgetting to check

Test your rule on the numbers you were given before using it for the 100th term.

Class activity 1 of 3

Pattern detective

Type each answer and press Check. Use the rule, not a long list.

Answers (for the teacher)
  1. 30
  2. 36
  3. 162
  4. 21
  5. 21
  6. 119
  7. 20
  8. 1275

Class activity 2 of 3

Matchstick houses

A row of houses is built from matchsticks: 1 house needs 6 sticks, 2 houses need 11, 3 houses need 16. Investigate the pattern.

  1. Draw 1, 2 and 3 houses and count the sticks to check 6, 11, 16.
  2. Find the gap between figures.
  3. Use jumps to find far terms.
  4. Work backwards for the last two.

Answers (for the teacher)
  1. 21
  2. 51
  3. 5
  4. 20
  5. 39

Class activity 3 of 3

The handshake investigation

Everyone in a room shakes hands with everyone else exactly once. 2 people make 1 handshake, 3 people make 3, 4 people make 6.

  1. Act it out with 4 people (or 4 coins) and count.
  2. Spot the pattern: 1, 3, 6, … are triangle numbers.
  3. Find the rule for n people: n × (n − 1) ÷ 2.
  4. Use it backwards for the last question.

Answers (for the teacher)
  1. 10
  2. 15
  3. 45
  4. 190
  5. 12

Quiz · 8 questions

Check what you learned

Pick an answer to see if it is right and why. Your best score is saved on this device.

1What comes next: 2, 6, 12, 20, 30, …?
Show the answer

42. The gaps are 4, 6, 8, 10, so the next gap is 12: 30 + 12 = 42. (These are 1×2, 2×3, 3×4, …; next is 6×7.)

2What is the 50th number in 7, 11, 15, 19, …?
Show the answer

203. 49 jumps of 4: 7 + 196 = 203.

3What is the 10th triangle number?
Show the answer

55. 10 × 11 ÷ 2 = 55.

4Matchstick squares in a row need 4, 7, 10, … sticks. How many sticks for 12 squares?
Show the answer

37. 4 + 11 × 3 = 37.

5What comes next: 1, 1, 2, 3, 5, 8, …?
Show the answer

13. Add the last two: 5 + 8 = 13.

6What comes next: 3, 6, 12, 24, …?
Show the answer

48. Each number doubles.

7Which term of 4, 7, 10, … is 100?
Show the answer

33rd. (100 − 4) ÷ 3 = 32 jumps, so the 33rd term.

81 + 2 + 3 + … + 20 = ?
Show the answer

210. 20 × 21 ÷ 2 = 210.

After class · project

My pattern puzzle book

Invent a growing pattern of shapes (dots, matchsticks, tiles) and turn it into three puzzles for someone at home.

What to make: A page with your four drawings, the rule, the 10th and 100th figures, and three puzzles with answers.

Stretch: Make a pattern whose gaps grow (like 1, 3, 6, 10) and find its 10th figure.

How it is marked

Look forDone well when
DrawingsFour figures that follow one clear rule
RuleThe rule is written in words and works for every figure
Far terms10th and 100th are correct with working
PuzzlesThree puzzles with correct answers

Project template

Copy it onto paper or print the practice sheet.

Figure 1: ___ pieces   Figure 2: ___   Figure 3: ___   Figure 4: ___
Gap between figures: ___   Rule in words: ______________________
Figure 10: ___   Figure 100: ___   (show the working)
Puzzle 1: ______________________   Answer: ___
Puzzle 2: ______________________   Answer: ___
Puzzle 3: ______________________   Answer: ___

Homework

Practise at home

Try each on paper, then tap to see the answer and working. Or print the practice sheet.

  1. Write the first six square numbers.

    1, 4, 9, 16, 25, 36

  2. Find the 100th number in 2, 5, 8, 11, …

    299 (2 + 99 × 3)

  3. What comes next: 81, 27, 9, 3, …?

    1 (divide by 3 each time)

  4. How many dots are in the 8th square figure?

    64

  5. A pattern starts 10, 17, 24, …. Is 101 in it?

    Yes. 101 − 10 = 91 and 91 ÷ 7 = 13 exactly, so 101 is the 14th term.

For parents and teachers

The 60-minute lesson plan

Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.

TimePartThe teacherThe learner
0–4 minWarm-upAsk the warm-up question and wait for a guess.Guess, then reason.
4–12 minDifferencesTeach, show the examples step by step, then the practice.Practise up to the stretch question.
12–21 minShapesTeach, use the interactive part, then the practice.Practise; try the stretch.
21–28 minDoubling and FibonacciTeach and show the examples.Practise.
28–35 minWorking backwardsTeach and show the examples.Practise; the stretch can be homework.
35–42 minActivity 1Let the learner work alone; hint only after a wrong check.Solve and check.
42–49 minActivity 2Follow the steps together for the first item.Finish the investigation.
49–55 minActivity 3Set as homework if time is short.Solve the puzzles.
55–59 minQuizRun the quiz and review mistakes.Answer and read why.
59–60 minSet the projectShow the template and the due date.Start the project.

Questions to ask

  • How do you know whether a pattern adds or multiplies?
  • Why is the 20th term 19 jumps from the first?
  • How can two triangles of dots make a rectangle?

If it feels hard

Build every pattern with coins first and count, before using any rule.

If it feels easy

Find the 100th number in 1, 3, 6, 10, … and the sum 1 + 3 + 5 + … + 99.

FAQ

Number patterns: find the rule: questions

What kind of pattern questions come in maths Olympiads?

Finding the next term, a far term like the 100th, which term a number is, and patterns made from shapes such as triangle and square numbers. This lesson covers all four.

What is the formula for the nth term of an adding pattern?

First term + (n − 1) × gap. For 5, 8, 11, … the 20th term is 5 + 19 × 3 = 62.

Which class is this Olympiad lesson for?

It suits Classes 4 to 7. Younger learners can use the visual and the first two parts; older learners should finish the challenge questions.

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