Patterns in Mathematics
20 Class 6 IMO Mathematics Olympiad practice questions on patterns in mathematics, part of the Algebraic Thinking topic area. Every question below includes the correct answer and a full explanation.
Practice this topic interactively →1. What is the 5th triangular number? (Triangular numbers: 1, 3, 6, 10, 15, ...)
- A.10
- B.12
- C.15
- D.21
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Answer: C. 15
Triangular numbers are 1, 3, 6, 10, 15, ... — the 5th one is 15.
2. Which of these sequences represents Virahānka numbers (each number is the sum of the two before it, starting 1, 2)?
- A.1, 2, 3, 5, 8, 13
- B.1, 2, 4, 8, 16
- C.1, 3, 5, 7, 9
- D.2, 4, 6, 8, 10
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Answer: A. 1, 2, 3, 5, 8, 13
Starting from 1 and 2: 1+2=3, 2+3=5, 3+5=8, 5+8=13 — giving 1, 2, 3, 5, 8, 13.
3. The figure shows a complete graph — every pair of points joined by a line. How many line segments are there?
- A.2
- B.3
- C.4
- D.6
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Answer: B. 3
With 3 points, every pair gets one segment: 3×2÷2 = 3.
4. How many line segments are there in a complete graph with 4 points, if every pair of points is joined by a segment?
- A.4
- B.5
- C.6
- D.8
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Answer: C. 6
Every pair of the 4 points gets one segment: 4×3÷2 = 6 segments.
5. Hexagonal numbers go 1, 7, 19, 37, ... (the gap between terms increases by 6 each time). What are the next two hexagonal numbers after 37?
- A.61 and 91
- B.55 and 73
- C.58 and 85
- D.64 and 97
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Answer: A. 61 and 91
Gaps go 6, 12, 18, 24, 30: 37+24=61, then 61+30=91.
6. Which of the following is a triangular number?
- A.20
- B.21
- C.22
- D.24
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Answer: B. 21
Triangular numbers: 1, 3, 6, 10, 15, 21, 28, ... — 21 is the 6th one.
7. The figure shows a complete graph with every pair of points joined. How many line segments are there?
- A.4
- B.5
- C.6
- D.8
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Answer: C. 6
With 4 points: 4×3÷2 = 6 segments.
8. How many line segments are in the complete graph shown?
- A.8
- B.9
- C.10
- D.12
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Answer: C. 10
With 5 points: 5×4÷2 = 10 segments.
9. Find the 8th triangular number using the rule: nth triangular number = n(n+1)/2.
- A.28
- B.32
- C.36
- D.40
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Answer: C. 36
8th triangular number = 8×9/2 = 36.
10. A complete graph has 21 line segments (every pair of points joined). How many points does it have?
- A.6
- B.7
- C.8
- D.9
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Answer: B. 7
If there are n points, the number of segments is n(n-1)/2. For n=7: 7×6/2 = 21.
11. The sum of the first 7 odd numbers is a perfect square. What is it? (1+3+5+7+9+11+13)
- A.36
- B.42
- C.49
- D.56
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Answer: C. 49
The sum of the first n odd numbers is always n². For n=7, that's 7² = 49.
12. A sequence of regular polygons starts with a square, and each next polygon has 2 more sides than the one before. What is the 4th shape in this sequence?
- A.Octagon
- B.Nonagon
- C.Decagon
- D.Dodecagon
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Answer: C. Decagon
Square (4 sides) → Hexagon (6) → Octagon (8) → Decagon (10). The 4th shape is a decagon.
13. The number of small squares in the nth stacked-square pattern is n². How many small squares are in the 7th stacked-square shape?
- A.36
- B.42
- C.49
- D.56
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Answer: C. 49
For n=7: 7² = 49.
14. The complete graph shown has 6 points, each joined to every other point. How many line segments does it have in total?
- A.12
- B.14
- C.15
- D.18
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Answer: C. 15
With 6 points: 6×5÷2 = 15 segments.
15. What is the 10th cube number in the sequence 1, 8, 27, 64, ...?
- A.900
- B.1000
- C.1100
- D.1210
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Answer: B. 1000
The nth cube number is n³. The 10th cube number is 10³ = 1000.
16. If the first two Virahānka numbers are 1 and 2, find the 9th Virahānka number. (1, 2, 3, 5, 8, 13, 21, 34, ...)
- A.34
- B.55
- C.89
- D.144
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Answer: B. 55
Continuing the sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55 — the 9th term is 55.
17. In a stacked-triangle pattern, the number of small triangles in the base row of shape 1, 2, 3, ... follows 1, 3, 5, 7, .... How many small triangles are in the base row of the 10th shape?
- A.17
- B.18
- C.19
- D.21
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Answer: C. 19
The base-row count follows 2n−1. For the 10th shape: 2×10−1 = 19.
18. The 5th and 6th triangular numbers are 15 and 21. Their sum equals a perfect square. What is it?
- A.30
- B.36
- C.42
- D.49
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Answer: B. 36
15 + 21 = 36 = 6². (In general, the sum of two consecutive triangular numbers is always a perfect square.)
19. Which of these numbers is NEITHER a triangular number NOR a Virahānka number? (Triangular: 1,3,6,10,15,21,... Virahānka: 1,2,3,5,8,13,21,...)
- A.10
- B.13
- C.16
- D.21
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Answer: C. 16
10 is triangular, 13 is a Virahānka number, and 21 is both — but 16 appears in neither list.
20. What is the total number of line segments in a complete graph with 9 points?
- A.28
- B.32
- C.36
- D.40
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Answer: C. 36
Number of segments = n(n-1)/2 = 9×8/2 = 36.