Number patterns: find the rule: practice sheet

Part 1: Look at the differences

Remember: 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.

1. 1. Warm-up Next number: 4, 9, 14, 19, …

2. 2. Warm-up Next number: 50, 46, 42, 38, …

3. 3. Core The 25th number in 3, 8, 13, …

4. 4. Core The 40th number in 100, 97, 94, …

5. 5. Stretch 🔥 A pattern adds the same amount each time. The 1st number is 7 and the 10th is 43. What is the gap?

6. 6. Stretch 🔥 How many numbers are in the list 5, 9, 13, …, 201?

Part 2: Growing shapes: triangle and square numbers

Remember: 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.

1. 7. Warm-up The 6th triangle number

2. 8. Warm-up The 12th square number

3. 9. Core 1 + 2 + 3 + … + 100 = ?

4. 10. Core Which triangle figure has exactly 45 dots? (the figure number)

5. 11. Stretch 🔥 The sum of the first 10 odd numbers, 1 + 3 + 5 + … + 19

6. 12. Stretch 🔥 How many dots are in figure 15 of the rectangle pattern n × (n + 1)?

Part 3: Doubling patterns and Fibonacci

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

1. 13. Warm-up Next number: 5, 10, 20, 40, …

2. 14. Warm-up Next number: 2, 2, 4, 6, 10, 16, …

3. 15. Core The 7th number in 1, 3, 9, 27, …

4. 16. Core The 10th Fibonacci number (1, 1, 2, 3, 5, …)

5. 17. Stretch 🔥 Next number: 1, 2, 4, 7, 11, 16, …

6. 18. Stretch 🔥 Next number: 2, 3, 5, 9, 17, …

Part 4: Working backwards: which term is it?

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

1. 19. Warm-up Which term of 2, 5, 8, … is 29?

2. 20. Core Which term of 7, 12, 17, … is 102?

3. 21. Core A row of triangles made of matchsticks needs 3, 5, 7, … sticks. How many sticks for 20 triangles?

4. 22. Stretch 🔥 Is 500 in the pattern 3, 7, 11, 15, …? (yes or no)

5. 23. Stretch 🔥 A staircase uses 1, 3, 6, 10, … blocks. How many blocks does a 12-step staircase need?

Activity 1: Pattern detective

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

1. 24. Next number: 2, 9, 16, 23, …

2. 25. Next number: 1, 4, 9, 16, 25, …

3. 26. Next number: 2, 6, 18, 54, …

4. 27. Next number: 1, 3, 6, 10, 15, …

5. 28. Next number: 2, 3, 5, 8, 13, …

6. 29. The 30th number in 3, 7, 11, 15, …

7. 30. Which term of 6, 11, 16, … is 101? (just the number)

8. 31. 1 + 2 + 3 + … + 50 = ?

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

1. 32. Sticks for 4 houses

2. 33. Sticks for 10 houses

3. 34. The gap (sticks added per house)

4. 35. How many houses use exactly 101 sticks?

5. 36. The most houses you can build with 200 sticks

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

1. 37. Handshakes with 5 people

2. 38. Handshakes with 6 people

3. 39. Handshakes with 10 people

4. 40. Handshakes with 20 people

5. 41. How many people make exactly 66 handshakes?

Homework

1. 42. Write the first six square numbers.

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

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

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

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

Project: My pattern puzzle book

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

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: ___