A Square and a Cube

Why This Matters

Here is a puzzle from the Ganita Prakash textbook.

A queen leaves her fortune of precious stones inside a safe. The code is hidden behind 100 lockers, numbered 1 to 100. A hundred people line up. Person 1 opens every locker. Person 2 toggles every 2nd locker — closes it if open, opens it if closed. Person 3 toggles every 3rd locker, Person 4 every 4th, and so on, up to Person 100.

At the end, only a few lockers are still open. They reveal the code. Which lockers stay open?

You could act it out, 100 lockers at a time. But there is a beautiful shortcut. A locker ends up open only if it was toggled an odd number of times. And how many times is a locker toggled? Exactly once for each of its factors. Locker 6 is touched by persons 1, 2, 3 and 6 — its four factors.

So the question becomes: which numbers from 1 to 100 have an odd number of factors? The answer turns out to be the square numbers — 1, 4, 9, 16, 25, … — and figuring out why is the start of this whole chapter.

This chapter is about squares and cubes: what they are, the secret patterns hidden inside them, and how to undo them with square roots and cube roots. By the end, the locker puzzle will feel obvious.

The Big Idea

A square number is what you get when you multiply a number by itself — like 5 × 5 = 25. A cube number is what you get when you multiply a number by itself three times — like 5 × 5 × 5 = 125. These two ideas come from real shapes: a square’s area is side × side, and a cube’s volume is side × side × side. Once you see squares and cubes as shapes made of unit pieces, their strange-looking patterns — odd factor counts, special last digits, sums of odd numbers — all start to make sense. And every square or cube can be undone: the square root and the cube root take you back to the side you started with.

Let’s Break It Down

We will go in order. First, square numbers and why they are called “squares”. Then the patterns hidden in perfect squares. Then square roots — how to undo a square. Then cube numbers, and finally cube roots.

Before we begin, let us refresh one idea the whole chapter leans on: factors and prime factorisation.

Square numbers and the area model

Back to the locker puzzle. We said a locker stays open only if it has an odd number of factors. Let us see why squares are the special numbers with an odd factor count.

Factors come in pairs. For 6, the pairs are 1 × 6 and 2 × 3. Each factor has a partner so that the two multiply back to 6. Because the partners are all different, the factors come in neat couples: 6 and 3 — that is 4 factors, an even number.

But look at 9. Its pairs are 1 × 9 and 3 × 3. Here something special happens: 3 partners with itself. We do not count that 3 twice — it is just one factor. So 9 has the factors 1, 3, 9 — that is 3 factors, an odd number.

This only happens when a number is a number times itself — that is, a square. The picture below shows both cases side by side.

Tap to zoom

So in the locker puzzle, every locker whose number is a square — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 — is toggled an odd number of times and stays open. There are exactly 10 of them.

Now, why do we even call 1, 4, 9, 16 “squares”? Because they are literally the areas of squares. A square with side 3 units is filled by 3 × 3 = 9 little unit squares. A square with side 4 holds 4 × 4 = 16. The number of unit squares (the area) is side × side. The figure below makes this clear.

Tap to zoom

We write this with a small raised 2. So 3 × 3 = 3² (read “three squared”), 4 × 4 = 4² = 16, and in general n × n = n². The squares of the natural numbers — 1, 4, 9, 16, 25, … — are called perfect squares.

You can square fractions and decimals too. For example (2.5)² = 2.5 × 2.5 = 6.25, and (³⁄₅)² = ³⁄₅ × ³⁄₅ = ⁹⁄₂₅. Squaring just means “multiply the thing by itself”, whatever the thing is.

Let’s use the factor idea on a slightly bigger number.

Patterns hidden in perfect squares

If you write out the first squares — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, … — they hide some neat patterns. These patterns are not just curiosities; they are quick tests you can use in exams.

Pattern 1 — the last digit. Look at the units digit (the last digit) of each square: 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, … Every perfect square ends in 0, 1, 4, 5, 6 or 9. None ever ends in 2, 3, 7 or 8.

Why? The last digit of a square depends only on the last digit of the number you squared. Square the digits 0 through 9 and look at their last digits: 0→0, 1→1, 2→4, 3→9, 4→6, 5→5, 6→6, 7→9, 8→4, 9→1. The results only ever end in 0, 1, 4, 5, 6 or 9. The digits 2, 3, 7, 8 simply never appear. So they can never be the last digit of a square.

This gives a one-glance test. The number 4,67,853 ends in 3, so it cannot be a perfect square. But be careful — the test only works one way.

Pattern 2 — zeros at the end. Square a number that ends in zeros and watch the zeros double. 10² = 100 (one zero → two zeros). 200² = 40000 (two zeros → four zeros). 700² = 490000 (two zeros → four zeros). So a square always has an even number of zeros at the end. A number ending in an odd number of zeros (like 1000, three zeros) can never be a perfect square.

Pattern 3 — parity. The square of an even number is even (2² = 4, 4² = 16). The square of an odd number is odd (3² = 9, 5² = 25). So a number and its square always have the same parity (both odd or both even).

Pattern 4 — squares are sums of odd numbers. This is the prettiest one. Look at the gaps between consecutive squares:

4 − 1 = 3

9 − 4 = 5

16 − 9 = 7

25 − 16 = 9

The gaps are the odd numbers 3, 5, 7, 9, … So if you add up odd numbers starting from 1, you keep landing exactly on the next square:

1 = 1 = 1²

1 + 3 = 4 = 2²

1 + 3 + 5 = 9 = 3²

1 + 3 + 5 + 7 = 16 = 4²

1 + 3 + 5 + 7 + 9 = 25 = 5²

This is not a coincidence — it has a lovely picture proof. Build a square out of dots, one inverted-L layer at a time. Each new L wraps around the previous square and adds the next odd number of dots. The figure below shows it.

Tap to zoom

This pattern is also a test: a number is a perfect square exactly when you can reach 0 by subtracting 1, 3, 5, 7, … in turn. Take 25: 25 − 1 = 24, 24 − 3 = 21, 21 − 5 = 16, 16 − 7 = 9, 9 − 9 = 0. We subtracted 5 odd numbers and hit 0, so 25 = 5². Try 38 and you cross past 0 without ever landing on it, so 38 is not a perfect square.

The pattern also lets you jump from one square to the next. The n-th odd number is 2n − 1 (1st is 2×1−1 = 1, 6th is 2×6−1 = 11, and so on). So to get from one square to the next, add the right odd number.

There is also a link to triangular numbers — 1, 3, 6, 10, 15, … (dots arranged in growing triangles). Add any two neighbouring triangular numbers and you get a perfect square: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16. Two triangles slot together into a square.

Square roots — undoing a square

Squaring takes a side and gives an area. The square root does the reverse: it takes the area and gives back the side.

Suppose a square garden has an area of 49 square metres. What is the length of its side? We need a number that, times itself, gives 49. Since 7 × 7 = 49, the side is 7 metres. We say 7 is the square root of 49, and write √49 = 7.

In general, if y = x², then x is a square root of y, written x = √y.

One subtlety: both 7 × 7 = 49 and (−7) × (−7) = 49 (a negative times a negative is positive). So a perfect square actually has two square roots, one positive and one negative: √49 = +7 or −7. In this chapter we will only use the positive square root.

So how do we test whether a big number like 324 is a perfect square, and find its root if it is? The most reliable method uses prime factorisation.

The idea: a perfect square is a number times itself. So in its prime factorisation, the primes must split into two identical groups. Whatever is in one group, multiplied out, is the square root. The figure below shows this for 324, and also shows a number (156) that fails the test.

Tap to zoom

Let’s run this test on a fresh number, step by step.

What if the number is not a perfect square, but you still want a rough idea of its root? Then you estimate by trapping it between two squares you know.

Let’s estimate the square root of a number that isn’t a perfect square.

Cube numbers — multiplying three times

Squares come from flat squares. Cubes come from solid cubes. A cube is a solid box where all edges are equal and all corners are right angles — like a dice or an ice cube.

How many tiny 1 cm cubes fit inside a cube of side 2 cm? Picture it as layers. The bottom layer is a 2 × 2 square of small cubes = 4 cubes. There are 2 such layers stacked up. So 2 × 2 × 2 = 8 small cubes. For a cube of side 3, each layer is 3 × 3 = 9 cubes, and there are 3 layers: 3 × 3 × 3 = 27. The figure below shows the idea.

Tap to zoom

We write a number times itself three times with a small raised 3. So 5 × 5 × 5 = 5³ (read “five cubed”) = 125, and in general n × n × n = n³. The numbers 1, 8, 27, 64, 125, … are called perfect cubes.

Is 9 a cube? No — 2³ = 8 and 3³ = 27, and there is nothing in between, so no number from 10 to 26 is a cube either. Cubes grow fast, so there are far fewer of them. Just like squares, you can cube fractions and negatives: (−6)³ = −6 × −6 × −6 = −216 (two negatives make a positive, then one more negative makes the answer negative).

Cubes hide patterns too. Add consecutive odd numbers in growing blocks and you get the cubes:

1 = 1³

3 + 5 = 8 = 2³

7 + 9 + 11 = 27 = 3³

13 + 15 + 17 + 19 = 64 = 4³

Block n uses n odd numbers, and they always add to n³.

There is a famous cube story. The mathematician Srinivasa Ramanujan was ill in hospital, and his friend G. H. Hardy came to visit. Hardy mentioned that his taxi was numbered 1729, a “rather dull number”. Ramanujan instantly replied that it was very interesting — it is the smallest number that can be written as the sum of two cubes in two different ways:

1729 = 1³ + 12³

1729 = 9³ + 10³

Because of this, 1729 is now called the Hardy-Ramanujan number, and such numbers are called taxicab numbers.

Cube roots — undoing a cube

Just as the square root undoes squaring, the cube root undoes cubing. Since 8 = 2³, we say 2 is the cube root of 8 and write ³√8 = 2.

In general, if y = x³, then x = ³√y.

To find a cube root, we use prime factorisation again — but with one change. For a square, we split the primes into two identical groups. For a cube, we split them into three identical groups. The figure below shows this for 3375, and shows 500 failing the test.

Tap to zoom

Why threes? Because cubing a number repeats every prime factor three times. If 15 = 3 × 5, then 15³ = (3 × 5)³ = 3 × 3 × 3 × 5 × 5 × 5 — each prime appears exactly three times. So a number is a perfect cube exactly when each of its primes appears a number of times that is a multiple of 3, letting them form complete triples.

Let’s find a cube root from scratch.

There is also a quick “guessing” trick for nice cube roots, using the last digit. The cubes of 0–9 end in 0,1,8,7,4,5,6,3,2,9 — and crucially each last digit is unique. So the last digit of a cube tells you the last digit of its cube root. For 1331: it ends in 1, and the only digit whose cube ends in 1 is 1, so the root ends in 1. Since 1331 is a 4-digit number between 10³ = 1000 and 20³ = 8000, the root is between 10 and 20. Ending in 1 → the root is 11. Indeed 11³ = 1331.

Common Mistakes

These trip up students every year. Spotting them now keeps the marks safe.

Quick Check

Answer each one, then read the explanation that appears.

Practice Problems

Try each one yourself first, then tap to see the full solution.

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Summary

• A number multiplied by itself is a square number; the squares of natural numbers (1, 4, 9, 16, …) are perfect squares. We write n × n as n².
• A number multiplied by itself three times is a cube number; 1, 8, 27, 64, … are perfect cubes. We write n × n × n as n³.
• Only square numbers have an odd number of factors, because in a square one factor pairs with itself.
• Perfect squares end only in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8, and have an even number of trailing zeros. (A number ending in an allowed digit still need not be a square.)
• The sum of the first n odd numbers is n² (a fact you can see by building a square out of inverted-L dot layers). Cubes are sums of blocks of consecutive odd numbers.
• The square root undoes a square; the cube root undoes a cube. The symbols are √ and ³√.
• A number is a perfect square if its prime factors split into two identical groups, and a perfect cube if they split into three identical groups. One whole group is the root.

What’s Next

You can now spot squares and cubes on sight, find their roots, and use their hidden patterns. That number-sense will keep paying off.

Next, in Chapter 2 — Power Play, you will go beyond squares and cubes to powers in general — what 2⁵ or 10⁸ really mean, the rules for multiplying and dividing powers, and how powers let us write enormous (and tiny) numbers neatly. The squaring and cubing you mastered here are just the first two steps of that bigger story. Onward!