Chapter 1: Patterns in Mathematics (Number and Shape Patterns)

• Shopping and money: adding prices, checking the change, finding discounts and comparing which pack is cheaper per kilogram.
• Cooking: measuring ingredients and doubling or halving a recipe for more or fewer people.
• Time and travel: reading clocks and calendars, planning when to leave, working out distances and fares.
• Home and school: measuring cloth or a room for tiles or paint, sharing things equally, scoring in games and sports.
• Saving: keeping a budget and working out how long it will take to save for something.

Shopping, cooking, measuring, telling time, planning travel, sharing, budgeting and playing games all use mathematics.

• Science: measuring and recording results of experiments, and finding the patterns in them (for example, the laws of motion and gravitation that let us send satellites and rockets into space).
• Economy and democracy: banking, taxes, budgets and trade use calculations; counting votes and planning elections need accurate counting and data.
• Buildings and bridges: engineers use geometry and measurement to design strong, safe structures and calculate the loads they can carry.
• Machines and technology: mobile phones, computers and TVs work using mathematics (codes, signals, algorithms); designing bicycles, cars, trains and planes needs measurement, speed and shape calculations.
• Calendars and clocks: they are based on the regular patterns in the motion of the Sun, Moon and Earth.

Mathematics underlies science, the economy and elections, construction, all modern machines and technology, and our calendars and clocks.

Yes. The rule for each sequence is given in the table in the next answer.

Yes: every sequence follows a rule (see the table in Q2).

See the table: e.g. triangular 36, 45, 55; squares 64, 81, 100; cubes 343, 512, 729; Virahānka 34, 55, 89; powers of 2: 128, 256, 512; powers of 3: 2187, 6561, 19683.

The next (6th) picture in each sequence:

• All 1's: a single dot.
• Counting numbers: a row of 6 dots.
• Odd numbers: 11 dots in an L-shape (6 dots across and 6 down, sharing the corner dot).
• Even numbers: 12 dots in two rows of 6.
• Triangular numbers: 21 dots in a triangle of 6 rows (1, 2, 3, 4, 5, 6).
• Squares: 36 dots in a 6 × 6 square.
• Cubes: 216 small cubes making a 6 × 6 × 6 cube.

All 1's: 1 dot; counting: 6 dots; odd: 11 dots (L-shape); even: 12 dots (2 rows of 6); triangular: 21 dots (6 rows); squares: 6 × 6 = 36; cubes: 6 × 6 × 6 = 216.

• Triangular numbers: that many dots can be arranged to form a triangle, with rows of 1, 2, 3, 4, … dots (e.g. 10 = 1 + 2 + 3 + 4).
• Square numbers: that many dots can be arranged in a square, with equal rows and columns (e.g. 16 = 4 rows of 4).
• Cubes: that many small cubes can be stacked to make a bigger cube, with equal length, breadth and height (e.g. 27 = 3 × 3 × 3).

Because those numbers of dots (or blocks) can be arranged exactly as a triangle, a square and a cube respectively.

Other examples: 1 is a triangle, a square and a cube; 6 can be a triangle (1 + 2 + 3) or a 2 × 3 rectangle; 12 can be a 3 × 4 rectangle, a 2 × 6 rectangle, or two rows of 6; 10 is a triangle (1 + 2 + 3 + 4) and also a 2 × 5 rectangle.

36 = 1 + 2 + … + 8 (a triangle of 8 rows) = 6 × 6 (a square).

Each new picture adds a ring of dots around the previous hexagon. The rings have 6, 12, 18, … dots (6 more each time):

$$1\stackrel{+6}{\to }7\stackrel{+12}{\to }19\stackrel{+18}{\to }37\stackrel{+24}{\to }61$$

The next hexagonal number is 37 + 24 = 61.

• Powers of 2: each picture is two copies of the previous one put side by side: 1 dot → 2 dots in a row → a 2 × 2 square (4) → two such squares stacked into a 2 × 2 × 2 cube (8) → two cubes side by side (16), and so on.
• Powers of 3: each picture is three copies of the previous one: 1 dot → 3 dots in a row → a 3 × 3 square (9) → three squares stacked into a 3 × 3 × 3 cube (27) → three such cubes in a row (81), and so on.

Another way is a branching tree: start with one dot and let every dot branch into 2 (or 3) new dots at the next level; the number of dots at each level gives 1, 2, 4, 8, … (or 1, 3, 9, 27, …).

Join 2 (or 3) copies of the previous picture each time: dot → row → square → cube → …, or draw a tree in which each dot branches into 2 (or 3).

Split a square grid of dots into L-shaped layers: the layers have 1, 3, 5, 7, … dots.

So the sum of the first $n$ odd numbers is an $n\times n$ square:

$$1+3+5+\cdots +19=10\times 10=100$$

$$1+3+5+\cdots +199=100\times 100=10000$$

First 10 odd numbers: 100. First 100 odd numbers: 10,000.

Take a square of dots and count it along slanting lines (diagonals). The slanting rows have 1, 2, 3, …, up to the side length, and then go back down to 1.

A 4 × 4 square has 16 dots, and its slanting rows give 1 + 2 + 3 + 4 + 3 + 2 + 1. The same works for a square of any size, so adding up to $n$ and back down gives $n\times n$.

Counting the dots of an n × n square along the slanting rows gives 1 + 2 + … + n + … + 2 + 1, so the sum is n × n.

Going up to 100 and back down is the slanting count of a 100 × 100 square:

$$1+2+\cdots +99+100+99+\cdots +2+1=100\times 100=10000$$

10,000

Adding up: 1, 1 + 1, 1 + 1 + 1, … = 1, 2, 3, 4, … , the counting numbers.

Adding up and down: "up and down" means taking 1 term, then 3 terms, then 5 terms, … (just as 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1 have 1, 3, 5 terms):

1, 1 + 1 + 1, 1 + 1 + 1 + 1 + 1, … = 1, 3, 5, 7, … , the odd numbers.

Up: counting numbers 1, 2, 3, 4, …; up and down: odd numbers 1, 3, 5, 7, …

1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, … = 1, 3, 6, 10, 15, … , the triangular numbers.

Picture: put rows of 1, 2, 3, 4, … dots one below the other, starting each row at the left edge. They form a right-angled "staircase" triangle. Each new row adds the next counting number, so the total is always a triangular number.

Triangular numbers 1, 3, 6, 10, 15, …: rows of 1, 2, 3, 4, … dots form a triangle.

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

Why: a square of dots can be cut along a slanting line, just above the diagonal, into two staircase triangles. One has rows of 1, 2, 3, 4 dots and the other has rows of 1, 2, 3; that is, two consecutive triangular numbers.

Square numbers 4, 9, 16, 25, …, because any square of dots splits into two consecutive triangular numbers.

Sums: 1, 3, 7, 15, 31, 63, …

Add 1: 2, 4, 8, 16, 32, 64, … : the powers of 2 again (each sum is one less than the next power of 2).

Why: start with the extra 1 and keep adding:

$$1+1=2,2+2=4,4+4=8,8+8=16,\dots$$

The extra 1 together with the first term makes 2, which together with the next term 2 makes 4, and so on. Each step doubles the total, so we always get the next power of 2.

The sums are 1, 3, 7, 15, 31, …; adding 1 gives 2, 4, 8, 16, 32, …, the powers of 2, because the 1 plus each power of 2 keeps doubling.

$$6\times 1+1=7,6\times 3+1=19,6\times 6+1=37,6\times 10+1=61,\dots$$

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

Picture: a hexagonal arrangement of dots can be divided into a centre dot and 6 equal triangles around it; each triangle is a triangular number of dots.

The hexagonal numbers 7, 19, 37, 61, 91, …, since a hexagon of dots is 1 centre dot plus 6 triangular groups.

1, 1 + 7 = 8, 8 + 19 = 27, 27 + 37 = 64, … : the cubes.

Picture: build a cube of small blocks layer by layer from one corner. A 1 × 1 × 1 cube is 1 block. To make a 2 × 2 × 2 cube, add 7 blocks that cover three faces of the small cube; to make 3 × 3 × 3, add 19 more blocks on three faces, and so on. Each added layer, seen from the corner, looks like a hexagon, and contains exactly a hexagonal number of blocks: 8 − 1 = 7, 27 − 8 = 19, 64 − 27 = 37.

The cubes 1, 8, 27, 64, …: each hexagonal number is a three-faced "shell" (hexagonal when seen from a corner) that grows a cube to the next size.

A few examples:

• Even numbers = 2 × counting numbers: 2 = 2 × 1, 4 = 2 × 2, … (two equal rows of dots).
• Even numbers = counting numbers + counting numbers: 1 + 1, 2 + 2, 3 + 3, …
• Sum of consecutive cubes is a square of a triangular number: 1 = 1², 1 + 8 = 9 = 3², 1 + 8 + 27 = 36 = 6², 1 + 8 + 27 + 64 = 100 = 10² (1, 3, 6, 10 are triangular numbers).
• Virahānka numbers: adding them up gives 2 less than a Virahānka number two places ahead: 1 + 2 + 3 + 5 = 11 = 13 − 2.
• Powers of 3: adding 1, 3, 9, 27 gives 40, and 2 × 40 + 1 = 81, the next power of 3.

For example: even numbers are twice the counting numbers; 1 + 8 + 27 + 64 = 100 = 10² (sums of cubes are squares of triangular numbers); sums of Virahānka numbers are 2 less than a later Virahānka number.

• Regular polygons: each shape has one more side (and corner) than the previous one: triangle, square, pentagon, … decagon; all sides and corners are equal.
• Complete graphs: $K_n$ has $n$ points, and every point is joined to every other point by a line.
• Stacked squares: an $n\times n$ arrangement of little squares, growing by one row and one column each time.
• Stacked triangles: a big triangle made of little triangles, with one more row each time.
• Koch snowflake: start with a triangle; each time, every line segment is replaced by a "speed bump" of 4 smaller segments.

Yes: polygons gain a side; complete graphs join every pair of points; stacked squares and triangles add a row each time; in the Koch snowflake every segment is replaced by a bump of 4 smaller segments.

• Regular polygons: next is an 11-sided regular polygon (hendecagon). Easy to draw roughly, though making all sides exactly equal needs care.
• Complete graphs: next is $K_7$: 7 points with every pair joined (21 lines).
• Stacked squares: next is a 6 × 6 arrangement (36 little squares).
• Stacked triangles: next has 6 rows (36 little triangles).
• Koch snowflake: the next shape is very hard to draw by hand, because each of the many already tiny segments must be replaced by an even tinier bump (the number of segments becomes 4 times as large).

The rules are as described in Q1.

All the next shapes can be drawn except, practically, the next Koch snowflake, whose segments become too tiny and numerous to draw by hand.

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

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

Why: in a closed shape made of straight sides, every side ends at a corner where it meets the next side, and every corner joins exactly two sides. Going around the shape, sides and corners come one after the other, so there are as many corners as sides.

Both give 3, 4, 5, 6, … (counting numbers from 3), because in a polygon each corner is where two neighbouring sides meet, so the numbers of sides and corners are equal.

$K_2,K_3,K_4,K_5,K_6$ have 1, 3, 6, 10, 15 lines: the triangular numbers.

Why: when we add a new point to a complete graph, it must be joined to every point already there. So:

• $K_2$: 1 line
• $K_3$: 1 + 2 = 3 lines
• $K_4$: 1 + 2 + 3 = 6 lines
• $K_5$: 1 + 2 + 3 + 4 = 10 lines

Adding counting numbers up gives triangular numbers.

1, 3, 6, 10, 15 (triangular numbers), because each new point adds as many new lines as there are points already.

1, 4, 9, 16, 25, the square numbers. The $n$-th shape has $n$ rows with $n$ little squares in each row, so it has $n\times n$ squares.

1, 4, 9, 16, 25 (square numbers), since the n-th shape has n rows of n squares.

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

Counting row by row from the top, the rows have 1, 3, 5, 7, … little triangles (each row has one more upright triangle and one more upside-down triangle than the row above). So the $n$-th shape has

$$1+3+5+\dots (n\text{ odd numbers})=n\times n$$

little triangles, since adding odd numbers gives square numbers.

1, 4, 9, 16, 25 (square numbers): the rows have 1, 3, 5, 7, … triangles, and the sum of the first n odd numbers is n × n.

The first shape (a triangle) has 3 segments. Each "speed bump" turns 1 segment into 4 smaller segments, so the number of segments is multiplied by 4 at each step:

$$3,3\times 4=12,12\times 4=48,48\times 4=192,192\times 4=768,\dots$$

This is 3 times the powers of 4: 3 × 1, 3 × 4, 3 × 16, 3 × 64, 3 × 256, …

3, 12, 48, 192, 768, … (3 × powers of 4), because each segment is replaced by 4 segments every time.