Square numbers
1, 4, 9, 16, 25, … are the square numbers. The square number in position n is n × n, written , and it is the number of tiles in a square with n tiles along each side.
Going from one square to the next adds a border of tiles along two sides. From 9 to 16 the border is 3 tiles, another 3 tiles and 1 in the corner, which is 7. So the gaps between the square numbers are the odd numbers 3, 5, 7, 9, …
The 4 by 4 square holds 16 tiles. The 3 by 3 square inside it holds 9, and the border around it adds 7.
3² = 9 < 16: a square of side 3 holds too few tiles, so √16 is more than 3
Grow the square until it holds 16 tiles: which side? √16 = ?
Grow the square one row and one column at a time. The number of tiles goes 1, 4, 9, 16 and 25: these are the square numbers.
Triangle numbers
1, 3, 6, 10, 15, … are the triangle numbers. The triangle number in position n is the number of dots in a staircase with 1 dot in its first column, 2 in the next, and so on up to n.
Each step adds one more dot than the step before: add 2, then 3, then 4. So the triangle number in position n is 1 + 2 + 3 + … + n, and the 5th is 10 + 5 = 15.
Each term is the one before with a new column one dot taller, so the gaps are 2, 3 and 4.
Two staircases make a rectangle
Take two copies of the same staircase and turn one upside down. They fit together into a rectangle n dots high and n + 1 dots wide, which holds n(n + 1) dots. One staircase is half of the rectangle, so the triangle number in position n is half of n(n + 1).
For n = 4 that is 4 × 5 = 20, and half of 20 is 10. For n = 20 it is 20 × 21 = 420, and half of 420 is 210, found without adding twenty numbers.
1 + 2 + … + n is a staircase; a second copy turned upside down fills its gaps
Set n = 6 and slide the second staircase in
Slide the second staircase into the first. Together they make a rectangle n by n + 1, so one staircase is half of n(n + 1).
The Fibonacci sequence
In the Fibonacci sequence each term is the sum of the two terms before it. It starts 1, 1, and then 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8 and 5 + 8 = 13.
The rule needs two terms to make the next one, so the sequence has to be given its first two terms before the rule can start.
Each jump along the line is the term two places back: from 8 to 13 the jump is 5.
The usual mistakes
Squaring is not doubling: the 6th square number is 6 × 6 = 36, not 6 × 2 = 12.
A triangle number is not a square number: the 6th triangle number is half of 6 × 7, which is 21, while the 6th square number is 36.
Fibonacci adds the two terms before, not the last term twice: after 5, 8 comes 5 + 8 = 13, not 8 + 8 = 16.
Worked example: Quadratic / Square Array Pattern
Question A pattern of square tile arrays is shown: Figure 1 has a total of 4 tiles (2 × 2). Figure 2 has 9 tiles (3 × 3). Figure 3 has 16 tiles (4 × 4). (a) How many tiles are in Figure 25? (b) Which figure number consists of exactly 1089 tiles?
1.Notice the grid dimensions: Fig 1 is 2 × 2, Fig 2 is 3 × 3, Fig 3 is 4 × 4.
Figure 1 is 2 × 2, Figure 2 is 3 × 3, Figure 3 is 4 × 4. 2.Pattern rule: The side length of the square is always (Figure Number + 1).
The side is always one more than the figure number. 3.(a) Figure 25 side length = 25 + 1 = 26. Total tiles = 26 × 26 = 676.
(a) Figure 25 has side 26: 26 × 26 = 676 tiles. 4.(b) To find the figure with 1089 tiles, find its side length: √1089 = 33.
(b) 1089 tiles is a 33 × 33 square. 5.Deduct 1 from the side length: 33 − 1 = 32 ⟹ Figure 32.
Side 33 means Figure 33 − 1 = 32.
Answer: (a) 676 tiles; (b) Figure 32
Common mistakes
- Squaring the figure number directly: 252 = 625 instead of (25+1)2 = 676.
- Confusing side length with figure number (stating Figure 33 instead of Figure 32).
More patterns and page numbers problems, worked step by step →
Worked example: Triangular Number Pattern (Staircase / Handshakes)
Question Dots are arranged in a triangular staircase formation: Figure 1 has 1 dot. Figure 2 has 3 dots (1 + 2). Figure 3 has 6 dots (1 + 2 + 3). Figure 4 has 10 dots (1 + 2 + 3 + 4). (a) How many dots are there in Figure 20? (b) If a figure has 120 dots, what is its figure number?
1.Join two identical Figure 1 triangles: forms a 1 × 2 rectangle of 2 dots ⟹ 2 ÷ 2 = 1.
Figure 1 with its mirror copy: a 1 × 2 rectangle, 2 dots, so the figure is 1. 2.Join two identical Figure 2 triangles: forms a 2 × 3 rectangle of 6 dots ⟹ 6 ÷ 2 = 3.
Figure 2 doubled is 2 × 3 = 6, so the figure is 3. 3.Join two identical Figure 3 triangles: forms a 3 × 4 rectangle of 12 dots ⟹ 12 ÷ 2 = 6.
Figure 3 doubled is 3 × 4 = 12, so the figure is 6. A figure and its copy always make n × (n + 1). 4.(a) For Figure 20: Double the dots to form a rectangle of 20 × 21 = 420. Divide by 2: 420 ÷ 2 = 210 dots.
(a) Figure 20 doubled: 20 × 21 = 420, so 210 dots. 5.(b) For 120 dots: Double the dots to find the rectangle: 120 × 2 = 240.
(b) 120 dots doubled is 240, a rectangle of two consecutive sides. 6.Find consecutive numbers that multiply to 240: 15 × 16 = 240. Figure number is the smaller number: 15.
15 × 16 = 240: the smaller side is the figure, 15.
Answer: (a) 210 dots; (b) Figure 15
Common mistakes
- Treating the sequence as linear and attempting to find a constant difference (3 − 1 = 2, 6 − 3 = 3).
- Forgetting to multiply by 2 before factoring for part (b) (e.g., trying to factor n(n+1) = 120 to get n ≈ 10.4).
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