Chapter 1: A Square and A Cube

Stack one row of $5$ tiles. Add another row of $5$ . Keep going until you have $5$ rows. You have built a $5 \times 5$ square holding $25$ tiles. Now imagine stacking five copies of that square on top of each other , you would get a perfect cube of $125$ tiles. The numbers $25$ and $125$ are not random; they belong to two of the most useful families in mathematics: squares and cubes.

Why care about these? Almost every formula in geometry, physics, and engineering hides a square or a cube somewhere. The area of a piece of land is measured in square metres. The volume of a water tank is measured in cubic metres. Doubling the side of a photograph makes its area $4$ times bigger, not $2$ . Doubling the side of an ice cube makes it $8$ times heavier. Squares and cubes are the secret reasons.

In this chapter you will learn how to recognise a perfect square or perfect cube on sight, find its square root or cube root, and use prime factorisation as a powerful microscope to see inside any number. You will also meet several patterns , like the sum of the first $n$ odd numbers being a square , that have entertained mathematicians for thousands of years, from ancient India to modern computer science.

By the end you will be comfortable with the symbols $\sqrt{\,}$ and $\sqrt[3]{\,}$ , you will know which numbers can never be perfect squares, and you will see why $1, 8, 27, 64, \dots$ keep showing up in surprising places.

What's inside

Square numbers and their patterns , what a perfect square is, and the patterns hidden inside the list $1, 4, 9, 16, \dots$
Square roots and how to find them , undoing a square, using prime factorisation and estimation.
Cube numbers and their patterns , perfect cubes, properties, and a famous taxi-cab story.
Cube roots , undoing a cube using prime factorisation.
Squares, cubes and the real world , area, volume, growth, and a few puzzles.

Key results

Idea	Symbol / formula	Example
Square of $n$	$n^2 = n \times n$	$7^2 = 49$
Cube of $n$	$n^3 = n \times n \times n$	$4^3 = 64$
Square root	$\sqrt{n^2} = n$ (for $n \ge 0$ )	$\sqrt{81} = 9$
Cube root	$\sqrt[3]{n^3} = n$	$\sqrt[3]{125} = 5$
Sum of first $n$ odd numbers	$1 + 3 + 5 + \dots + (2n-1) = n^2$	$1 + 3 + 5 + 7 = 16 = 4^2$
Square of $(n+1)$	$(n+1)^2 = n^2 + (2n+1)$	$11^2 = 100 + 21 = 121$
Difference of consecutive cubes	$(n+1)^3 - n^3 = 3n^2 + 3n + 1$	$5^3 - 4^3 = 75 + 15 + 1 = 91$

Last-digit rule. A perfect square can only end in $0, 1, 4, 5, 6, 9$ . So $1027$ can never be a square , it ends in $7$ . A perfect cube, however, can end in any digit.

How to read this chapter

Keep grid paper and a small box of cubes nearby. Squares and cubes are visual ideas; almost every result becomes obvious once you draw it. After each worked example, hide the answer and try it from scratch.

Chapter 1: A Square and A Cube

What's inside

Key results

How to read this chapter

Sub-topics

Practice quiz