1. Patterns in Mathematics

• Paying for fruits, vegetables and groceries, and counting the change.
• Reading the time and planning a timetable.
• Measuring cloth, length of a room or area of a plot.
• Cooking — measuring the amount of each ingredient.
• Finding the speed of a vehicle and the time for a journey.

• Science: scientists use mathematics to measure, record and study the results of experiments.
• Economy and democracy: budgets, taxes, banking and counting of votes use mathematics.
• Building: bridges, houses and dams are designed using measurements and calculations.
• Machines: TVs, mobile phones, computers, trains, cars and planes are designed using mathematics.
• Calendars and clocks: they are based on patterns in the motion of the sun, moon and earth.

Yes. Each sequence follows a rule. The rules are written in the answer to Question 2 below.

• All 1’s — one dot again.
• Counting numbers — a line of 6 dots.
• Odd numbers — 11 dots (an L-shape with 6 dots on each arm).
• Even numbers — 12 dots in two rows of 6.
• Triangular numbers — 21 dots in a triangle with rows of 1, 2, 3, 4, 5, 6 dots.
• Squares — 36 dots in a 6 × 6 square.
• Cubes — a cube made of 6 × 6 × 6 = 216 small cubes.

Triangular numbers: These many dots can be arranged in the shape of a triangle.

Squares: These many dots can be arranged in the shape of a square (1×1, 2×2, 3×3, …).

Cubes: These many small cubes can be arranged to make a bigger cube (1×1×1, 2×2×2, …).

Another example: 1 is also both triangular and square. 6 is both triangular and a rectangle (2 × 3).

These are hexagonal numbers.

Differences: 7 − 1 = 6, 19 − 7 = 12, 37 − 19 = 18. The next difference is 24.

Next number = 37 + 24 = 61

Powers of 2: Start with 1 dot. Each time, make a copy of the picture and place it beside the old one. We get 1, 2, 4, 8, 16 dots.

Powers of 3: Start with 1 dot. Each time, make 2 copies of the picture and place all three together, like a triangle. We get 1, 3, 9, 27 dots.

The sum of the first 10 odd numbers makes a 10 × 10 square.

1 + 3 + 5 + … + 19 = 10 × 10 = 100

The sum of the first 100 odd numbers makes a 100 × 100 square.

1 + 3 + 5 + … + 199 = 100 × 100 = 10,000

Draw a square of dots and tilt it so that it stands on one corner (like a diamond).

Now count the dots row by row from top to bottom.

For a 4 × 4 square, the rows have 1, 2, 3, 4, 3, 2, 1 dots.

So 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4 × 4.

Adding up to 100 and back down gives a 100 × 100 square.

Value = 100 × 100 = 10,000

Up: 1, 1 + 1 = 2, 1 + 1 + 1 = 3, … We get the counting numbers 1, 2, 3, 4, …

Up and down: 1, 1 + 1 + 1 = 3, 1 + 1 + 1 + 1 + 1 = 5, … We get the odd numbers 1, 3, 5, 7, …

1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, … We get the triangular numbers.

Picture: Put 1 dot in the first row, 2 dots in the second row, 3 dots in the third row, and so on. The dots form a triangle.

1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, … We get the square numbers.

Why: Draw a square of dots and cut it along a diagonal. One part is a triangle with one row more than the other. For 4 × 4: 10 dots + 6 dots = 16.

Sums: 1, 3, 7, 15, 31, 63, …

Adding 1 to each: 2, 4, 8, 16, 32, 64, … These are the powers of 2 again.

Why: 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16, … Each time, the extra 1 doubles the total and makes the next power of 2.

1 × 6 + 1 = 7, 3 × 6 + 1 = 19, 6 × 6 + 1 = 37, 10 × 6 + 1 = 61, 15 × 6 + 1 = 91, …

We get the hexagonal numbers 7, 19, 37, 61, 91, …

Picture: A hexagon of dots has one dot in the centre. The remaining dots split into 6 equal triangles.

1, 8, 27, 64, … We get the cubes.

Picture: Look at a cube from one corner. It looks like a hexagon. A cube can be built layer by layer, and each new outer layer seen from the corner is a hexagonal number.

• The differences between consecutive squares are the odd numbers: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7.
• Adding two consecutive counting numbers gives odd numbers: 1 + 2 = 3, 2 + 3 = 5, 3 + 4 = 7.
• Each Virahānka number is the sum of the previous two numbers.
• Adding up the even numbers: 2, 2 + 4 = 6, 2 + 4 + 6 = 12, … gives 2 times the triangular numbers.

• Regular Polygons: each shape has one more side than the previous shape (3, 4, 5, …, 10 sides).
• Complete Graphs: each shape has one more point, and every point is joined to every other point.
• Stacked Squares: each shape is a bigger square grid (1×1, 2×2, 3×3, …).
• Stacked Triangles: each shape has one more row of small triangles.
• Koch Snowflake: each line segment is replaced by a ‘speed bump’ to get the next shape.

• Regular Polygons: next is a regular polygon with 11 equal sides (hendecagon). Rule: add one side.
• Complete Graphs: next is K7 — 7 points, each joined to the other 6 (21 lines).
• Stacked Squares: next is a 7 × 7 grid of 49 small squares.
• Stacked Triangles: next has 6 rows with 36 small triangles.
• Koch Snowflake: replace every small segment with a speed bump. It becomes very hard to draw because the segments become very tiny.

Sides: 3, 4, 5, 6, 7, 8, 9, 10 — counting numbers starting from 3.

Corners: 3, 4, 5, 6, 7, 8, 9, 10 — the same sequence.

Why: In a closed shape, each side ends at a corner, and each corner joins two sides. So the number of sides is equal to the number of corners.

K2 = 1, K3 = 3, K4 = 6, K5 = 10, K6 = 15. These are the triangular numbers.

Why: When we add a new point, it is joined to all the old points. So we add 1, then 2, then 3, then 4, then 5 lines.

1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, …

1, 4, 9, 16, 25, 36. These are the square numbers.

Why: Each shape has the same number of rows and columns, so the count is 1 × 1, 2 × 2, 3 × 3, and so on.

1, 4, 9, 16, 25. These are the square numbers.

Why: The rows have 1, 3, 5, 7, 9, … triangles. Adding odd numbers gives square numbers.

For example, 1 + 3 + 5 = 9.

Each segment is replaced by 4 smaller segments, so the count is multiplied by 4 every time.

3, 3 × 4 = 12, 12 × 4 = 48, 48 × 4 = 192, 192 × 4 = 768, …

The sequence is 3 times Powers of 4: 3, 12, 48, 192, 768, …