Patterns in Mathematics
Chapter 1 — Full Solutions
Every Math Talk prompt and every Figure It Out question from Chapter 1, answered step‑by‑step with pictures, in‑text reasoning and clean worked numericals.
In‑Text Questions (Math Talk)
Quick prompts asked in the middle of the chapter, right where the idea comes up.
Yes — it happens forever, for every single odd number you add.
$1 + 3 + 5 + \cdots + (2n-1) = n^2$ — the sum of the first $n$ odd numbers is always the $n^{\text{th}}$ square number.
Split the square into L‑shaped layers (called “gnomons”). The innermost layer is a single dot (1). Wrapping an L‑shaped border of dots around it adds 3 dots, then the next border adds 5, then 7, and so on.
Because every square (of any size) can be peeled into these L‑shaped odd‑numbered layers, adding consecutive odd numbers always rebuilds a bigger square.
$1+3+5+7+9+11+13+15+17+19 = 10^2 = 100$
$1+3+5+\cdots+199 = 100^2 = 10{,}000$
Yes. Arrange the dots as a tilted diamond (square rotated 45°) instead of a grid — row by row it grows 1, 2, 3, … up to the middle and then shrinks 3, 2, 1 back down. Counted straight down the middle, that diamond is exactly an $n \times n$ square viewed at an angle.
$1+2+\cdots+(n-1)+n+(n-1)+\cdots+2+1 = n^2$
Exercise Questions (Figure It Out)
Every “Figure It Out” box from the chapter, section by section.
1.1 What is Mathematics? — Figure It Out
Page 2
Mathematics quietly runs almost everything we do in a day:
- Shopping: adding prices, calculating discounts, and checking change.
- Cooking: measuring ingredients and scaling a recipe up or down.
- Travel: reading a clock, estimating travel time, and calculating vehicle speed.
- Home & construction: finding the area of a room or plot before buying tiles or paint.
- Design: the repeating patterns seen in floor tiles, fabric prints, and building facades.
- Sports: keeping scores, calculating averages, and judging angles and distances.
(This is an open, discussion‑based question — the goal is to notice mathematics in ordinary moments, not to find one “correct” list.)
- Science: mathematics is the language used to record, compare, and verify the results of every scientific experiment.
- Economy & democracy: budgets, taxes, GDP, and even fair vote‑counting all depend on arithmetic and statistics.
- Construction: bridges and houses stay standing because engineers use geometry and calculations of force and load.
- Technology: mobile phones, computers, and TVs run on binary number systems and algorithms built from mathematical logic.
- Transport: the shapes of cars, trains, and planes are optimised using mathematical models of motion and air resistance.
- Timekeeping: calendars and clocks are built entirely on counting cycles — days, months, and years.
(Class discussion question — encourage students to connect each example back to a specific branch of mathematics.)
1.2 Patterns in Numbers — Figure It Out
Page 3 · Table 1
Yes — every sequence in Table 1 is built using one clear rule:
- Powers of 2: $1,\ 2,\ 4=2\times2,\ 8=2\times2\times2,\ 16=2\times2\times2\times2,\ldots$ — each term is double the one before it.
- Powers of 3: $1,\ 3,\ 9=3\times3,\ 27=3\times3\times3,\ldots$ — each term is triple the one before it.
- Virahānka numbers: $1,\,2,\,3,\,5=2+3,\,8=3+5,\,13=5+8,\ldots$ — each term is the sum of the two terms before it.
The rest of the sequences are explained fully with pictures in Table 2 — see the diagram below.
| Sequence | Next 3 terms | Rule |
|---|---|---|
| All 1’s | 1, 1, 1 | Every term is 1. |
| Counting numbers | 8, 9, 10 | Add 1 to the previous term. |
| Odd numbers | 15, 17, 19 | Add 2 to the previous term (starting at 1). |
| Even numbers | 16, 18, 20 | Add 2 to the previous term (starting at 2). |
| Triangular numbers | 36, 45, 55 | Add the next counting number ($+7,+8,+9,\ldots$). |
| Squares | 64, 81, 100 | $n \times n$ for $n = 8, 9, 10$. |
| Cubes | 343, 512, 729 | $n \times n \times n$ for $n = 7, 8, 9$. |
| Virahānka numbers | 34, 55, 89 | Add the two previous terms. |
| Powers of 2 | 128, 256, 512 | Multiply the previous term by 2. |
| Powers of 3 | 2187, 6561, 19683 | Multiply the previous term by 3. |
1.3 Visualising Number Sequences — Figure It Out
Page 5 · Table 2
The 6th picture in each row simply continues the same rule one more step:
| Sequence | 6th picture (value) |
|---|---|
| All 1’s | 1 dot |
| Counting numbers | 6 dots in a row |
| Odd numbers | 11 dots |
| Even numbers | 12 dots |
| Triangular numbers | 21 dots (a 6‑row triangle) |
| Squares | 36 dots (a 6×6 grid) |
| Cubes | 216 unit cubes (a 6×6×6 cube) |
- Triangular numbers: that many dots can always be arranged into a perfect equilateral triangle, with 1 dot in the top row and one extra dot in each row below it.
- Square numbers: that many dots can always be arranged into a perfect $n \times n$ square grid.
- Cubes: that many unit cubes can always be stacked into a perfect $n \times n \times n$ cube.
36 dots can be arranged perfectly as a triangle with 8 rows ($1+2+3+4+5+6+7+8=36$), and as a $6\times6$ square ($6\times6=36$).
This shows that one number can play two different geometric “roles” depending on how you arrange it — other such numbers exist too (e.g., try 1, and look up more triangular‑square numbers such as 1225).
These are called hexagonal numbers — the dots form rings around a central dot, building up a hexagon shape.
Next hexagonal number $= 61$
Powers of 2 can be shown as a point, then a line (2 points), a square (4 points), a cube (8 points), and a 4‑D hypercube sketch (16 points) — each shape is two copies of the previous one joined together.
Powers of 3 can be shown similarly using three copies joined together at each step: a point, a triangle (3 points), 3 triangles joined (9 points), and so on.
1.4 Relations among Number Sequences — Figure It Out
Page 8–9
Draw the dots as a diamond (a square tilted 45°). Each row of the diamond has $1,2,3,\ldots,n,\ldots,3,2,1$ dots — exactly the “up and down” sum — and together they fill an $n\times n$ square turned on its corner.
$1+2+\cdots+100+\cdots+2+1 = 100^2 = 10{,}000$
Adding the All 1’s sequence “up” gives the Counting numbers.
Adding the All 1’s sequence “up and down” gives $1, 3, 5, 7,\ldots$ — the Odd numbers.
$1,\ 1+2,\ 1+2+3,\ 1+2+3+4,\ldots = 1, 3, 6, 10,\ldots$ — this is the Triangular number sequence.
Picture it as a staircase: row 1 has 1 dot, row 2 has 2 dots, row 3 has 3 dots, and so on — stacking these staircase rows builds the triangle shape.
$1+3=4,\ \ 3+6=9,\ \ 6+10=16,\ \ 10+15=25,\ldots$ — these are the Square numbers.
Two consecutive triangular numbers are literally two triangles that lock together edge‑to‑edge to form one bigger square — nothing is left over and nothing overlaps.
$1+2+4+\cdots+2^{n-1} = 2^n – 1$, so adding 1 always gives $2^n$.
This happens because doubling and adding 1 is exactly what happens when you go from one power of 2 to the next — each new block is bigger than the sum of every block that came before it, by exactly 1.
$(1\times6)+1=7,\ \ (3\times6)+1=19,\ \ (6\times6)+1=37,\ \ (10\times6)+1=61,\ \ (15\times6)+1=91,\ldots$
This gives $7, 19, 37, 61, 91,\ldots$ — the Hexagonal numbers.
$1,\ 1+7=8,\ 1+7+19=27,\ 1+7+19+37=64,\ldots$
This gives $1, 8, 27, 64,\ldots$ — the Cube numbers!
A cube of side $n$ can be peeled into hexagonal layers around its centre, exactly like a square peels into odd‑numbered L‑shaped layers — so summing hexagonal numbers rebuilds a cube.
One nice example: adding up the cubes gives the square of a triangular number.
$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1+2+3+\cdots+n)^2$
Check it: $1^3+2^3+3^3 = 1+8+27 = 36$, and the triangular number $T_3 = 1+2+3 = 6$, and indeed $6^2 = 36$. ✓
(This is open‑ended — any correctly verified pattern the student discovers is a valid answer.)
1.5 Patterns in Shapes — Figure It Out
Page 11 · Table 3
- Regular Polygons: the number of sides increases by 1 each time — triangle (3), quadrilateral (4), pentagon (5) … decagon (10).
- Complete Graphs: each new figure adds one more point, joined by a line to every existing point.
- Stacked Squares: each figure is a bigger square grid, one row and one column larger than the last.
- Stacked Triangles: each figure adds one more row of small triangles along the base.
- Koch Snowflake: every straight edge is replaced by a “speed bump” of 4 smaller edges.
| Sequence | Next shape |
|---|---|
| Regular Polygons | Hendecagon — an 11‑sided regular polygon |
| Complete Graphs | $K_7$ — 7 points, each joined to all 6 others (21 lines) |
| Stacked Squares | A $6\times6$ grid of unit squares |
| Stacked Triangles | A triangle with 6 rows of small triangles |
| Koch Snowflake | Every edge again replaced with a smaller “speed bump” — a snowflake with $3\times4^{4}=768$ segments |
1.6 Relation to Number Sequences — Figure It Out
Page 11–12
Sides: $3, 4, 5, 6, 7, 8, 9, 10,\ldots$ — Counting numbers starting from 3.
Corners: $3, 4, 5, 6, 7, 8, 9, 10,\ldots$ — the exact same sequence.
They match because in any closed shape made only of straight edges, every side must start and end at a corner — so the number of sides always equals the number of corners.
$K_2, K_3, K_4, K_5, K_6$ have $1, 3, 6, 10, 15$ lines — the Triangular numbers.
Each time a new point is added, it must connect to every point already there — so the number of new lines equals the number of existing points, exactly the way triangular numbers grow (add 1, then 2, then 3, …).
$1, 4, 9, 16, 25,\ldots$ — the Square numbers.
The $n^{\text{th}}$ shape is an $n \times n$ grid, which by definition contains $n \times n = n^2$ little squares.
This gives $1, 4, 9, 16, 25,\ldots$ — the Square numbers once again.
$3,\ 3\times4=12,\ 12\times4=48,\ 48\times4=192,\ldots$ — the sequence is $3, 12, 48, 192, 768,\ldots$ i.e. $3 \times 4^{\,n-1}$ (3 times the Powers of 4).