Class 8 Mathematics Chapter 1: A Square and a Cube

A will with a puzzle in it

Queen Ratnamanjuri leaves her stones to whoever solves her puzzle first. A hundred people face a hundred lockers, all shut. The first person walks along opening each one. The second person then changes every 2 toggles every second one, person 3 every third one, and so on to the hundredth. Only some lockers end up open, and those reveal the code. Her son works out the answer before anyone touches a locker.

Who touches a locker

Locker 6 is touched by persons 1, 2, 3 and 6, and by nobody else. Those are exactly the factors of 6. A locker is therefore toggled once for each factor it has. It starts closed, so it finishes open only on an odd tally.

Factors come in pairs

Each factor has a partner, and the pair multiplies to the number. For 6 the pairs are 1 with 6, and 2 with 3. Four factors, so locker 6 closes. Pairing like this means most numbers have an even count of factors, which is why most of the hundred lockers end the way they started.

Where the pairing breaks

For 36 the pairs are 1 with 36, 2 with 18, 3 with 12, 4 with 9, and then 6 with 6. That last pair is one factor, not two, because both members are the same. The count comes out odd. A number whose factor list does this is a number that can be written as something times itself.

So the open lockers are the squares

A number built by multiplying some number by itself is a square. Only squares have an odd number of factors, so only the square-numbered lockers stay open: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Ten lockers out of a hundred. The puzzle is answered without opening anything.

Lockers touched exactly twice

The next clue asks for the lockers touched exactly twice. Two factors means 1 and the number itself and nothing between, which is what a prime number is. So the first five are 2, 3, 5, 7 and 11. One idea about factor counts answers both halves of the puzzle.

Why the name is square

A square tile of side 3 units holds 3 rows of 3 unit squares, so its area is 9. The area of a square of side n is n times n, and we write that as n squared. The name comes from the shape, and the squares of the counting numbers are called perfect squares.

Sides need not be whole

A square tile can have a side of 2.5 units, and its area is 2.5 times 2.5, which is 6.25. A side of three fifths gives an area of nine twenty-fifths. Squaring works on fractions and decimals exactly as it does on whole numbers, so the notation carries over unchanged.

The last digit rules some out

Write out the squares up to 30 squared and look at the final digits. Every one ends in 0, 1, 4, 5, 6 or 9. Not one ends in 2, 3, 7 or 8. So a number ending in 2, 3, 7 or 8 is certainly not a square, and that is a check worth doing first because it costs one glance.

But it proves nothing either way

The rule only works in one direction. Both 16 and 36 are squares ending in 6, and 26 ends in 6 and is not a square. A last digit of 0, 1, 4, 5, 6 or 9 leaves the question open. Ruling out is cheap; proving needs real work.

Zeros double, parity holds

A number ending in three zeros has a square ending in six, because the zeros are counted twice over. So a square can only carry an even number of trailing zeros. Parity carries over too: an even number has an even square and an odd number has an odd square, every time.

The gaps are the odd numbers

Subtract each square from the next: 4 minus 1 is 3, 9 minus 4 is 5, 16 minus 9 is 7, 25 minus 16 is 9. The gaps are the odd numbers in order. Read the other way, adding the odd numbers from 1 builds the squares one after another, which is the same fact seen from the other side.

Building a square from odd numbers

So the first n odd numbers add to n squared. The nth odd number is 2n minus 1, which gives the next one to add without counting along the list. Given that 35 squared is 1225, the 36th odd number is 71, and 1225 plus 71 is 1296. That is 36 squared, found by one addition.

Testing by taking odd numbers away

Run it backwards to test a number. From 25 take 1, then 3, then 5, then 7, then 9, and you land exactly on 0 at the fifth step: 25 is 5 squared. Do the same to 38 and it slips past 0 into a negative, so 38 cannot be built that way and is not a square.

How far apart the squares sit

Between 9 and 16 lie 10, 11, 12, 13, 14 and 15, which is six numbers. Between 16 and 25 lie eight. The count between n squared and the next square is 2n, so the squares thin out as the numbers grow. Ten of them fit below 100, and only four more between 101 and 200.

Triangles hiding in the squares

The triangular numbers are 1, 3, 6, 10, 15, built by adding 1, then 2, then 3, and so on. Add two that sit next to each other: 1 and 3 give 4, 3 and 6 give 9, 6 and 10 give 16. Two triangles of consecutive size always fit together into a square.

Scenario:

A school is paving a square courtyard with square tiles. Each tile is one metre across. The store has 1,089 tiles. The caretaker wants to know whether they will make an exact square with none left over. He also asks how many tiles a courtyard one metre wider on each side would take. The art teacher adds a second question. The border tiles are to be painted. So the count of tiles ADDED when the side grows by one matters for the paint order too, not only for the tiling.

Analysis:

The question is whether 1,089 is a perfect square, and then how the count changes when the side grows by one. Both are answered by the odd-number pattern rather than by trial.

Title:

Hunt the odd factor count

Safety:

No equipment is needed, so there is nothing to handle carefully.

What You Notice:

Every number you marked is a square, and none of the others is. The odd count and the square are the same thing seen two ways.

The hundred-locker puzzle has the same answer for any number of lockers: the open ones are always the squares.

Ten of the numbers from 1 to 100 are squares, but only four more turn up between 101 and 200. Squares thin out fast.

Rule out, never prove — Endings 2, 3, 7 and 8 kill a square at once. The other six endings settle nothing at all.

Undoing a square

A square tile has an area of 49 square units. What is its side? We need a number that gives 49 when multiplied by itself, and 7 times 7 is 49. So the side is 7 units. So 7 is one square root of 49. Put generally, when y equals x times x, we call x a square root of y.

Two roots, not one

What is the square root of 64? Eight times eight is 64, so 8 is one answer. But minus 8 times minus 8 is also 64. Every perfect square therefore has two integer square roots, one positive and one negative. Both are correct answers to the question as asked.

What the radical sign means

The radical sign is written for the positive root only. So the sign in front of 64 names 8, not minus 8. Throughout this chapter we take the positive root, because a side length or a tile count cannot be negative. The negative root still exists and still squares to the number.

Listing squares until you arrive

Is 576 a perfect square? One route is to write the squares in order until you reach it or pass it. From 20 squared the list runs 400, 441, 484, 529 and then 576. So 576 is 24 squared. The method is sound and gets slow as the numbers grow.

Subtracting odd numbers

A second route uses the odd-number pattern. From 81 take 1, then 3, then 5 and on. The totals fall 80, 77, 72, 65, 56, 45, 32, 17 and finally 0 at the ninth step. Landing exactly on zero proves 81 is a square, and the nine steps name its root as 9.

The useful route: prime factors

The third route is the one worth learning. A square is a number times itself, so its prime factors should divide into two identical groups. If they do, the product of one group is the square root. If some prime is left over, the number is not a square at all.

Working through 324

Break 324 into primes: 2 times 2 times 3 times 3 times 3 times 3. Those can be arranged as 2 times 3 times 3, twice over. So 324 is 18 times 18, which makes it 18 squared and gives a root of 18. The same factors can be read as the pairs 2 with 2, 3 with 3, and 3 with 3.

When the pairing fails

Now try 156. Its prime factorisation is 2 times 2 times 3 times 13. The two twos pair up, but the 3 and the 13 have no partners. The factors cannot be split into two identical groups, so 156 is not a perfect square. One unpaired prime is enough to settle it.

Making a number into a square

This also answers a harder question. What is the smallest number that 9408 must be multiplied by to make a square? Factorise it, find the primes that are left unpaired, and multiply by exactly those. Every prime then appears an even number of times, and the result is a square.

Trapping a root between squares

For a big number, start by trapping it. To find the root of 1936, note that 1936 lies between 1600 and 2500. Those are 40 squared and 50 squared, so the root lies between 40 and 50. We have narrowed fifty possibilities down to ten in one step.

Using the last digit

1936 ends in 6, and only numbers ending in 4 or 6 have squares ending in 6. So the root is 44 or 46. Two candidates remain out of the original fifty, and the last digit did that work for free.

Halving the interval

Test the middle. 45 squared can be worked out as 40 plus 5, all squared, which is 1600 plus 400 plus 25, or 2025. That is more than 1936, so the root is below 45. Of our two candidates only 44 is left, and 44 squared is indeed 1936.

Estimating a root that does not exist

Not every number has a whole-number root. Aribam says 250 and Bijou cannot answer, because 250 is not a square. But he can estimate. 250 lies between 225 and 256, which are 15 squared and 16 squared, so the root lies between 15 and 16. It sits much nearer 16.

A handkerchief from a square of cloth

Akhil has a square of cloth of area 125 square centimetres. Can he cut a square handkerchief of side 15? No, because 15 squared is 225, which is more cloth than he has. The nearest squares below and above are 121 and 144, so the largest whole-number side he can cut is 11 centimetres.

Reading a root off a known square

A known square gives its neighbours cheaply. Given that 125 squared is 15625, what is 126 squared? Add the 126th odd number, which is 251. So 126 squared is 15876. The same idea that built the squares now moves between them in either direction.

What each method is for

The last digit rules a square out in a glance. Subtracting odd numbers proves a square and names its root, but slowly. Prime factorisation does both and scales. Trapping between known squares gives a usable estimate when there is no whole-number root at all.

Scenario:

A park committee has 2,025 square paving slabs, each one metre square. They want a square lawn edged with those slabs and no slab left over. The treasurer asks whether the slabs make an exact square. The gardener wants the side length so she can order edging rope. A member suggests buying 50 more slabs to make the lawn bigger. The committee needs to know whether the new total would still be a perfect square, because a lawn that is not square cannot be edged the way they planned.

Analysis:

The first question is a squareness test, the second asks for the root, and the third tests whether a number near a square is also a square. Prime factorisation answers all three.

Title:

Trap a root in three steps

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

Three cheap steps usually settle a four-digit root. Trapping cuts the field, the last digit halves it, and one squaring finishes it.

The Babylonians kept clay tablets of squares and cubes around 1700 BCE so that roots could be looked up rather than worked out.

The word root comes from the Sanskrit mula, meaning the root of a plant, and the idea travelled into Arabic and Latin as jidhr and radix.

Split in two, then multiply one — Break the number into primes, split them into two identical groups, and the product of ONE group is the root.

From the solid to the number

A cube is a solid whose edges are all equal and whose faces meet at right angles. How many unit cubes make a cube of edge 2? Eight. Of edge 3? Twenty-seven. The counts 1, 8, 27 come from multiplying an edge by itself three times, and we call such numbers cubes.

Counting the layers

Take an edge of 4 units. Each flat layer is a 4 by 4 square, so it holds 16 unit cubes. There are 4 such layers stacked up, so the solid holds 4 times 16, which is 64. The same count is 4 times 4 times 4, and we write it as 4 cubed.

The notation

For any number n, the product n times n times n is written n cubed. So 5 cubed is 125, and the numbers 1, 8, 27, 64 and 125 are the first perfect cubes. Is 9 a cube? No. Two cubed is 8 and three cubed is 27, so nothing between them is a cube, 9 included.

Cubes of other kinds of number

Cubing works on any number, not only whole ones. Four sixths cubed is 64 over 216, because top and bottom are each cubed. 13.08 cubed is 2237.810112. And minus 6 cubed is minus 216, because three negative factors leave the sign negative.

Last digits behave differently

Squares end only in 0, 1, 4, 5, 6 or 9. Cubes are not so fussy: look down a table of cubes and every digit from 0 to 9 turns up as a last digit. So the last-digit test that rules out squares does not rule out cubes at all.

Zeros come in threes

Can a cube end in exactly two zeros? No. A number ending in one zero has a cube ending in three, because the zero is counted three times over. Trailing zeros on a cube therefore arrive three at a time, so a count of two is impossible.

A dull taxicab number

Hardy visited Ramanujan in hospital and mentioned that his taxi was numbered 1729, which he thought rather dull. Ramanujan replied at once. It is the least number reachable as a total of two cubes in two separate ways: 1 cubed plus 12 cubed, and 9 cubed plus 10 cubed.

Why that answer was possible

Numbers with that property are now called taxicab numbers, and the next two are 4,104 and 13,832. Ramanujan knew 1729 because he had spent years tinkering with numbers. His colleague Littlewood said that every positive integer seemed to be one of his personal friends.

Cubes from odd numbers

Odd numbers build cubes too, in runs. One is 1 cubed. Three plus five is 8, which is 2 cubed. Seven plus nine plus eleven is 27. Thirteen through nineteen gives 64. Each cube is a run of consecutive odd numbers, and the run for n cubed is n numbers long.

Reading a long run quickly

Later in the same pattern comes the run 91, 93, 95, 97, 99, 101, 103, 105, 107 and 109. There are ten numbers in it, so the total must be 10 cubed, which is 1,000. The pattern gives the answer without adding anything.

Cube roots

Since 8 is 2 cubed, the number 2 is said to be the cube root of 8. When y equals x cubed, we call x the cube root of y. So the cube root of 27 is 3 and the cube root of 1,000 is 10. The notation puts a small 3 against the radical sign.

Testing a cube with prime factors

The square test split the primes into two identical groups. For a cube, split them into three. Take 3,375: its primes are 3, 3, 3, 5, 5 and 5. Three groups of 3 times 5 each give 15, so 3,375 is 15 cubed and its cube root is 15.

When the triplets fail

Now take 500, which is 2 times 2 times 5 times 5 times 5. The three fives make a triplet, but the two twos cannot. No three identical groups can be formed, so 500 is not a perfect cube. The same single-leftover rule as before decides it.

Every prime three times over

Look at the pattern the other way. 15 is 3 times 5, and 15 cubed is 3, 3, 3, 5, 5, 5. 12 is 2 times 2 times 3, and 12 cubed carries each of those three times. A cube therefore carries every prime of its original three times over.

Where the words came from

The earliest known tables of squares and cubes were Babylonian clay tablets from about 1700 BCE, used for land measurement and building. In India the term varga meant both a square figure and the square power, while ghana meant both the solid cube and the third power. Both were in use by the third century BCE.

Why a root is called a root

The Sanskrit word mula means the root of a plant, and also a basis or origin. So varga-mula was the square root and ghana-mula the cube root. That image travelled onward: Arabic took jidhr and Latin took radix, both meaning the root of a plant. Brahmagupta wrote in 628 CE that the root of a square is simply the number it was squared from.

Scenario:

A sweet shop packs laddus into cubical boxes. Each box holds unit cells in a cube arrangement, so a box of edge 4 cells holds 64 laddus. The owner has an order for 3,375 laddus and wants one exact cubical box with no cell empty. His nephew suggests 500 laddus for a smaller order in the same style. The owner also wants to know how many laddus a box one cell wider on every edge would take, because the printer charges by the size of the label.

Analysis:

The first two questions are cube tests, and prime factorisation answers both. The third asks for the difference between two consecutive cubes, which is a subtraction once the edges are known.

Title:

Build cubes from odd runs

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

Each run adds to the next cube, and a run of n numbers gives n cubed. The cubes are hiding in the odd numbers.

Hardy thought 1729 a dull taxicab number. Ramanujan answered at once that 1,729 is the least number reachable as two cubes added, in two separate ways.

The Sanskrit word for the fourth power was varga-varga, the square of the square, which is exactly what it is.

Two groups square, three groups cube — Split the primes into two identical groups for a square and into three for a cube. One group is the root either way.