A Square and A Cube | CBSE Class 8 Maths Notes

This note covers square numbers, the locker puzzle, patterns in squares, odd and triangular numbers, square roots, prime factorisation, estimation, cube numbers, cube roots, taxicab numbers, successive differences and the history of squares and cubes.

How does the locker puzzle reveal square numbers?

In the locker puzzle, 100 people take turns changing the state of 100 lockers. To toggle a locker means to open it if closed and close it if open. Person 1 opens every locker; person 2 toggles every second locker.

Person 3 toggles every third locker, and the process continues until person 100 has taken a turn. Whole numbers are zero and the counting numbers; a positive number is greater than zero. A factor of a positive whole number divides it exactly, without a remainder. A locker is toggled by precisely those people whose numbers are its factors.

Property: Square numbers have an odd number of factors

An even number is divisible by 2; an odd number is not. Starting closed, a locker ends open after an odd number of toggles. After an even number of toggles, it ends closed.

Locker 6 is toggled by people 1, 2, 3 and 6. Its factors form partner pairs, meaning pairs whose product, or multiplication result, is the original number: 1 × 6 and 2 × 3. Here × means multiplication, and = means equality.

A square number is the product of a number with itself. For 36, the factor pair 6 × 6 repeats the same factor. This factor is counted once in the list of factors, while the remaining factors have distinct partners.

This explains why positive square numbers have an odd number of factors. The lockers remaining open are therefore 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Identifying the factor pattern replaces the need to follow every toggle separately.

Which lockers are touched exactly twice?

A prime number is a natural number with exactly two distinct factors, 1 and itself. Natural numbers are the counting numbers beginning with 1. Thus the first five lockers touched exactly twice are 2, 3, 5, 7 and 11, giving the passcode 2-3-5-7-11.

What do squares and perfect squares mean?

Definition: Squaring means multiplying a number by itself. If n represents any number, then n² = n × n. The raised 2 indicates two equal factors; read n² as “n squared”.

A geometric square has four equal sides and four right angles. A right angle is a quarter-turn angle. Its area, the amount of surface it covers, is its side length multiplied by itself. A unit square has sides of one unit, so it measures one square unit of area.

In this table, “sq.” abbreviates “square”. The numerical area of each square is a square number. The side length and the area describe different quantities, so the unit attached to an answer matters.

The drawing is a square divided into five rows and five columns of equal small squares. It shows 25 unit squares arranged as a larger square.

Reference: NCERT Class 8, page 3, unnumbered

How are perfect squares distinguished?

The squares of natural numbers are called perfect squares. Examples are 1, 4, 9, 16 and 25. This expression refers to the counting-number sequence, although the operation of squaring also applies to other kinds of numbers.

A fraction expresses a number as one quantity divided by another; the slash in 3/5 means division by 5. A decimal, such as 2.5, uses place values after a decimal point. Both fractions and decimals can be squared.

For a square of side 3/5 units, the area is (3/5)² = (3/5) × (3/5) = 9/25 square units. For a square of side 2.5 units, the area is (2.5)² = 2.5 × 2.5 = 6.25 square units. Parentheses group the number being squared.

What do the last digit, zeroes and parity tell us?

The units digit is the final digit of a whole number. All perfect squares end in 0, 1, 4, 5, 6 or 9. None end in 2, 3, 7 or 8. This provides a quick way to reject some numbers.

Property: A possible last digit does not prove a square

A number ending in one of the allowed digits is not necessarily a perfect square. Both 16 and 36 are squares, but 26 also ends in 6 and is not a square. The last-digit rule gives a test for impossibility, not a complete test for membership.

Worked example 1. What can the units digits tell us about 327 and 576?

Answer: 327 ends in 7, so it is not a perfect square. The final digit 6 in 576 is possible for a square, but that alone is inconclusive. Calculating 24² = 576 confirms that 576 is a perfect square.

Numbers ending in 1 or 9 have squares ending in 1. Squares such as 4² = 16 and 6² = 36 show that numbers ending in 4 or 6 have squares ending in 6. These patterns can help narrow a search for a square root.

How do trailing zeroes behave?

Trailing zeroes are zeroes at the end of a number. Squaring doubles their number. For example, 10² = 100 and 40² = 1600, while 100² = 10000 and 700² = 490000. A number with three trailing zeroes has six in its square.

Thus perfect squares can have only an even number of trailing zeroes. This rule concerns the end of the number, not zeroes occurring between other digits. Count the complete final run of zeroes when applying it.

Parity means whether a number is even or odd. Squaring preserves parity: an even number has an even square, and an odd number has an odd square. Last digits, trailing zeroes and parity give useful checks on a calculation.

How are perfect squares connected with consecutive odd numbers?

Consecutive numbers follow one another without skipping a member of the sequence. Consecutive odd numbers are 1, 3, 5, 7 and so on. Here “and so on” means that the same pattern continues.

The differences between successive squares are odd numbers: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7 and 25 − 16 = 9. The symbol − means subtraction. Each difference gives the amount needed to reach the next square.

Property: The first n odd numbers add to n²

A sum is the result of addition. Here n is the number of odd terms being added, starting at 1. For example, 1 + 3 + 5 + 7 + 9 = 25 = 5². The symbol + means addition. Every perfect square can be built from such an initial sequence.

Black dots form square arrays, with red inverted L-shaped boundaries marking additions. The labels develop from 1 + 3 to 1 + 3 + 5 and then 1 + 3 + 5 + 7.

Reference: NCERT Class 8, page 6, unnumbered

The added border extends the existing square in two directions. The diagram makes the odd-number additions visible, while the sums record the same growth numerically. It is a visual proof, an argument presented through a picture.

Worked example 2. Given 35² = 1225, find 36² using the odd-number pattern.

Answer: The nth odd number is 2n − 1, where n gives its position and 2n means 2 × n. The 36th odd number is 2 × 36 − 1 = 71. Therefore 36² = 1225 + 71 = 1296.

How does repeated subtraction test a square?

Worked example 3. Test whether 25 is a perfect square by subtracting successive odd numbers.

Answer: 25 − 1 = 24; 24 − 3 = 21; 21 − 5 = 16; 16 − 7 = 9; 9 − 9 = 0. Zero is reached after five subtractions, so 25 = 5².

For 38, the same process leaves 2 after subtracting 11, and subtracting the next odd number, 13, gives −11. This passes below zero without reaching it. Therefore 38 cannot be expressed as an initial sum of consecutive odd numbers and is not a perfect square.

How do triangular numbers and successive differences relate to squares?

Triangular numbers count dots arranged in triangular rows. The first illustrated values are 1, 3, 6, 10 and 15. Two consecutive triangular arrangements can be fitted together to make a square arrangement.

The examples are 1 + 3 = 4 = 2², 3 + 6 = 9 = 3² and 6 + 10 = 16 = 4². Each pair uses neighbouring triangular numbers. The addition joins their dots into a complete square.

Triangular dot groups are labelled 1, 3, 6, 10 and 15. Below them, outlined squares show the sums 1 + 3, 3 + 6 and 6 + 10, with their square-number totals.

Reference: NCERT Class 8, page 7, unnumbered

What are successive differences?

Successive differences are obtained by subtracting each term from the next, then repeating this operation on the resulting list. For squares, the first level gives consecutive odd numbers. Subtracting neighbouring values again gives a constant second level.

These levels describe different calculations. The first row contains square numbers. The next contains gaps between them. The final row contains gaps between those gaps. Keeping the rows separate helps explain why the second level becomes constant.

To count whole numbers strictly between two consecutive perfect squares, subtract the smaller square from the larger and then subtract 1. The final subtraction excludes the endpoint. For the squares of 16 and 17 there are 32 intervening numbers; for 99 and 100 there are 198.

What is a square root, and which root do we use?

A square root of a number is a number whose square equals it. Taking a square root reverses squaring, so these are inverse operations, operations that undo one another. If x is a number and y = x², then x is a square root of y.

A square with area 49 square centimetres has side length 7 centimetres because 7 × 7 = 49. The abbreviation cm means centimetre, and cm² means square centimetre. Side length uses a length unit; area uses a square unit.

Why do positive perfect squares have two integer roots?

Integers include zero, positive whole numbers and their negatives. A negative number is less than zero. Both 8 × 8 and (−8) × (−8) equal 64, so the two integer square roots of 64 are 8 and −8.

The symbol √ denotes the positive square root for the positive numbers considered here. Thus √64 = 8 and √100 = 10. When identifying both integer roots, say “8 and −8” rather than confusing the pair with the single value denoted by √64.

Note: The square-root calculations here use positive roots. A side length of a square is positive, so the area problem with 49 cm² gives 7 cm as its answer.

What methods can find a square root?

One method is to list squares until the target is reached or passed. For 576, the sequence 20² = 400, 21² = 441, 22² = 484, 23² = 529 and 24² = 576 reaches the required number.

Another method subtracts consecutive odd numbers from the target. For 81, zero is reached at the ninth subtraction, giving √81 = 9. Listing squares becomes inefficient for larger numbers; repeated subtraction for 729 is possible but time-consuming. Prime factorisation provides another method.

How does prime factorisation identify perfect squares?

Prime factorisation writes a natural number greater than 1 as a product of prime numbers. A prime factor is a factor that is prime. Repeated prime factors must all be included because their number matters when forming equal groups.

Property: Prime factors of a square form pairs

A number is a perfect square when its prime factors can be split into two identical groups. The product in either group is its positive square root. Equivalently, all occurrences of each prime factor can be arranged in pairs of equal factors.

Worked example 4. Is 324 a perfect square? Find its positive square root.

Answer: 324 = 2 × 2 × 3 × 3 × 3 × 3. Rearrange this as (2 × 3 × 3) × (2 × 3 × 3) = 18 × 18. Therefore 324 = 18² and √324 = 18.

The same factorisation can be displayed as (2 × 2) × (3 × 3) × (3 × 3). Selecting one factor from each pair gives 2 × 3 × 3. This explains why pairing and making two identical groups are equivalent procedures.

1. Write the complete prime factorisation of the number.
2. Arrange each repeated prime into pairs of equal factors.
3. Check whether any prime factor remains without a partner.
4. If all factors are paired, multiply one factor from each pair to obtain the positive square root.

Worked example 5. Is 156 a perfect square?

Answer: 156 = 2 × 2 × 3 × 13. The two factors of 2 form a pair, but 3 and 13 do not have matching partners. The complete factorisation cannot form two identical groups, so 156 is not a perfect square.

Having one available pair is insufficient. Every prime factor must be accounted for. In 156, grouping 3 with 13 would not repair the problem because a pair for this test consists of equal factors, not merely any two factors placed together.

How can square roots be estimated and checked?

To estimate a square root is to locate it approximately using familiar squares. An interval is the range between two bounds. The symbols < and > mean “less than” and “greater than”. Narrowing an interval makes an estimate more useful.

Worked example 6. Find √1936 by narrowing the possible interval.

Answer: 40² = 1600 and 50² = 2500, so 40 < √1936 < 50. Its square ends in 6, so a whole-number root must end in 4 or 6, giving candidates 44 and 46.

Since 45² = 2025 is greater than 1936, the root is below 45. The remaining candidate is 44. Checking 44 × 44 = 1936 confirms √1936 = 44.

The last-digit pattern narrows the candidates, but the final multiplication confirms the exact result. A candidate is a possible answer awaiting verification. This is especially important when it has not already been established that the target number is a perfect square.

What if the number is not a perfect square?

For 250, the broad bounds are 100 < 250 < 400, giving 10 < √250 < 20. Using 15² = 225 and 16² = 256 narrows the result to 15 < √250 < 16.

Since 256 is much closer to 250 than 225, √250 is approximately 16. It is still less than 16. An approximation is a nearby value, rather than an assertion that the number has exactly that square root.

Worked example 7. Akhil has a square cloth of area 125 cm². Can he cut a square handkerchief of side 15 cm? What is the largest integer side length possible?

Answer: 15² = 225, so a 15 cm square requires more than 125 cm². Since 11² = 121 and 12² = 144, the cloth's side lies between 11 cm and 12 cm. The largest integer side length is 11 cm.

In this problem, the condition “integer side length” determines the required answer. The cloth's actual side lies between two integers, but the requested handkerchief must use a whole-number length that fits within it.

What are cube numbers, and how can we picture them?

A geometric cube is a solid with equal edges meeting at right angles. A face is a flat surface of a solid. An edge is a line where two faces meet. A unit cube has an edge length of one unit. Equal unit cubes can build a larger cube.

Definition: The cube of a number is the product of three equal factors. If n denotes the number, n³ = n × n × n. The raised 3 means that n occurs three times as a factor.

The first positive perfect cubes, cubes of natural numbers, are 1, 8 and 27. They arise from 1 × 1 × 1, 2 × 2 × 2 and 3 × 3 × 3. A cube of side 2 cm contains eight cubes of side 1 cm.

Two cube drawings show grids of smaller cubes, with three divisions along an edge in one and four in the other. The larger example has four square layers of 16 unit cubes each.

Reference: NCERT Class 8, page 12, unnumbered

For edge length 4 units, each layer contains 4 × 4 = 16 unit cubes. There are four such layers, so the total is 4 × 4 × 4 = 64. Counting a single layer gives a square; counting all the layers gives a cube.

How do cubes differ from squares?

Squaring uses two equal factors, whereas cubing uses three. Neither operation means multiplying the original number by the raised digit. The notation describes repeated factors, and writing out the product helps prevent this confusion.

Not every natural number is a perfect cube. Since 2³ = 8 and 3³ = 27, 9 is not a perfect cube, and neither is any number from 10 to 26. These numbers lie between consecutive perfect cubes.

Cubes can end in any digit, and cubing preserves parity: an odd number has an odd cube and an even number has an even cube. Cubing triples the number of trailing zeroes. Therefore a perfect cube cannot end with exactly two zeroes.

Cubing also applies to fractions, decimals and negative numbers. The examples (4/6)³ = 64/216, (13.08)³ = 2237.810112 and (−6)³ = −216 show that the operation is not restricted to positive whole numbers.

Which patterns connect cubes with odd numbers and taxicab numbers?

Consecutive odd numbers also form cubes, but their grouping differs from the square pattern. For a square, the sum begins at 1 each time. For the cube pattern below, move through the odd numbers in successive groups containing one, two, three and then more terms.

Later in this pattern, 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109 equals 10³ = 1000. Recognising the appropriate group gives the sum without adding each term separately.

What makes 1729 special?

A taxicab number can be expressed as a sum of two positive cubes in two different ways. The number 1729 is the smallest such number and is known as the Hardy-Ramanujan number. Its two expressions are 1729 = 1³ + 12³ = 9³ + 10³.

G. H. Hardy visited Srinivasa Ramanujan in hospital after travelling in a taxicab numbered 1729. Hardy considered the number dull, but Ramanujan immediately identified its special cube property. The next two taxicab numbers given in this sequence are 4104 and 13832.

Can successive differences be explored for cubes too?

Begin with the sequence 1, 8, 27, 64, 125 and 216. Subtract neighbouring terms to make a new row, then repeat on that row. The first differences are 7, 19, 37, 61 and 91. The second differences are 12, 18, 24 and 30. The third differences are 6, 6 and 6. For this cube sequence, equal differences appear at the third level.

How do cube roots and prime factorisation work together?

A cube root of a number is a number that gives it when cubed. The symbol ∛ denotes cube root. Thus ∛8 = 2 because 2³ = 8. If x is a number and y = x³, then x = ∛y.

Property: Prime factors of a cube form triplets

A triplet here means three equal prime factors. A positive natural number greater than 1 is a perfect cube if its prime factors can be split into three identical groups. Equivalently, each prime's occurrences can be grouped into complete triplets.

Worked example 8. Determine whether 3375 is a perfect cube and find its cube root.

Answer: 3375 = 3 × 3 × 3 × 5 × 5 × 5. This becomes (3 × 5) × (3 × 5) × (3 × 5) = 15³. Therefore 3375 is a perfect cube and ∛3375 = 15.

Alternatively, the factors form one triplet of 3 and one triplet of 5. Taking one member from each triplet gives 3 × 5 = 15. The two arrangements explain the same cube-root calculation.

1. Find the complete prime factorisation of the number.
2. Collect equal prime factors into groups of three.
3. Check that no occurrence is left outside a complete triplet.
4. Multiply one factor from each triplet to obtain the cube root.

Worked example 9. Is 500 a perfect cube?

Answer: 500 = 2 × 2 × 5 × 5 × 5. The factors of 5 form a triplet, but the two factors of 2 do not. The factors cannot make three identical groups, so 500 is not a perfect cube.

The square and cube tests use the same factorisation but different grouping requirements. For the square test, every prime factor must belong to a complete pair of equal factors; for the cube test, every prime factor must belong to a complete triplet of equal factors. In both methods, an unmatched factor prevents the required grouping.

Useful cube-root results include ∛64 = 4, ∛512 = 8 and ∛729 = 9. Larger examples are ∛27000 = 30 and ∛10648 = 22. Each answer names the number whose cube is the given quantity.

Where do the ideas and names of squares and roots come from?

The first known list of perfect squares and perfect cubes was compiled by the Babylonians as far back as 1700 BCE. BCE means “Before Common Era”; CE means “Common Era”. The lists survive on clay tablets.

They were used to find square roots and cube roots quickly in calculations involving land measurement, architectural design and other geometric problems. “First known” refers to the surviving knowledge of these lists, rather than making a claim about every earlier calculation.

What did the Sanskrit terms mean?

A power records repeated equal factors, as in a square or cube. Varga referred both to a square figure or its area and to the square power. Ghana referred both to a solid cube and to a product of three equal factors. The fourth power was called varga-varga. These terms were used in India at least from the third century BCE.

Aryabhata, in 499 CE, connected varga with a square figure, its area and the product of two equal quantities. This connects the numerical operation with its geometric picture.

The Sanskrit word mula meant a plant's root, basis, cause or origin. Varga-mula meant square root and ghana-mula meant cube root. The mathematical use of mula is found in India at least from the first century BCE.

Arabic and Latin subsequently used their words for a plant's root, jidhr and radix, for the mathematical idea. Another Sanskrit term was pada, meaning foot, basis, cause or origin. Brahmagupta, in 628 CE, explained a square's root as that of which it is the square.

Glossary

• Square number — A number obtained by multiplying a number by itself as two equal factors.
• Perfect square — The square of a natural number, belonging to the sequence beginning 1, 4, 9 and 16.
• Factor — A positive whole number that divides another positive whole number exactly, without leaving a remainder.
• Prime number — A natural number having exactly two distinct factors, namely 1 and the number itself.
• Prime factorisation — Writing a natural number greater than 1 as a product made entirely of prime factors.
• Parity — The classification of a whole number according to whether it is even or odd.
• Trailing zeroes — The consecutive zeroes appearing at the end of a whole number's written form.
• Triangular number — A number that counts dots arranged in triangular rows, such as 1, 3, 6 or 10.
• Square root — A number which, when multiplied by itself, produces the original number under consideration.
• Perfect cube — A number obtained by multiplying a natural number by itself as three equal factors.
• Cube root — A number which produces the original number when used as three equal factors in multiplication.
• Triplet of prime factors — A group of three equal prime factors used when testing cubes or finding cube roots.
• Successive differences — Differences found between neighbouring terms, with the same subtraction process repeated on each resulting sequence.
• Taxicab number — A number expressible as a sum of two positive cubes in two different ways.

Common errors and misconceptions

• Misconception: A number ending in 6 must be a perfect square. Correct: 26 ends in 6 but is not a square. An allowed last digit does not establish the result.
• Misconception: Every number has an even number of factors. Correct: Positive perfect squares have an odd number because their equal-factor pair contributes one distinct factor.
• Misconception: Any sum of consecutive odd numbers is the square pattern. Correct: The square pattern uses every odd number starting from 1 up to the chosen endpoint.
• Misconception: Finding one pair of prime factors proves a perfect square. Correct: Every prime factor must be paired; 156 has a pair of 2s but leaves 3 and 13 unmatched.
• Misconception: √250 equals 16 exactly. Correct: Its value is between 15 and 16, and approximately 16, while remaining less than 16.
• Misconception: Cubing a number means multiplying it by 3. Correct: Cubing uses three equal factors, so 4³ = 4 × 4 × 4 = 64.
• Misconception: Pairs of prime factors are enough to identify a cube. Correct: Cubes require complete triplets, or three identical groups; 500 fails this condition.

Exam-style questions with model answers

Q1. Explain why 327 is not a perfect square and why the final digit of 576 is insufficient to establish that it is a square. [2 marks]

1. 327 ends in 7, but a perfect square cannot have 7 as its units digit.
2. 576 ends in 6, an allowed digit. This does not prove it is a square; a further test is required.

Q2. Given 35² = 1225, find 36² using the sum of consecutive odd numbers. Explain the steps. [3 marks]

1. The square 35² is the sum of the first 35 odd numbers beginning at 1. To obtain the next square, add the next odd number.
2. The 36th odd number is 2 × 36 − 1 = 71, using the position of the required term.
3. Therefore 36² = 1225 + 71 = 1296. This extends the sum from 35 odd terms to 36 odd terms.

Q3. Use prime factorisation to determine whether 324 is a perfect square and find its positive square root. [3 marks]

1. The prime factorisation is 324 = 2 × 2 × 3 × 3 × 3 × 3. Every factor in this expression is prime.
2. Arrange these factors into two identical groups: 324 = (2 × 3 × 3) × (2 × 3 × 3). Each group has product 18.
3. Thus 324 = 18², establishing that it is a perfect square. Its positive square root is √324 = 18.

Q4. Akhil has a square piece of cloth of area 125 cm². Decide whether he can cut a square handkerchief of side 15 cm, and find the largest possible integer side length. [3 marks]

1. A square handkerchief of side 15 cm would have area 15² = 225 cm². This exceeds the available 125 cm², so it cannot be cut from the cloth.
2. The neighbouring squares are 11² = 121 and 12² = 144. Hence the cloth's side is greater than 11 cm but less than 12 cm.
3. The largest integer side length that fits is therefore 11 cm. A side length of 12 cm would require too much area.

Q5. There are 100 initially closed lockers numbered 1 to 100. Each person numbered 1 to 100 toggles every locker whose number is divisible by their own, opening a closed locker or closing an open one. Explain which lockers remain open, and list them all. [5 marks]

1. A locker is toggled once for each factor of its number, because exactly those numbered people act on it.
2. Starting from closed, an odd number of toggles leaves a locker open. An even number leaves it closed.
3. Factors normally occur in distinct partner pairs whose product is the locker number. A square has one pair with equal members.
4. That repeated member is counted once as a distinct factor. Consequently, positive squares have an odd number of factors and their lockers remain open.
5. The complete list of open lockers is 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

Q6. Using prime factorisation, show that 3375 is a perfect cube and find its cube root. [3 marks]

1. Write the prime factorisation as 3375 = 3 × 3 × 3 × 5 × 5 × 5. There are three factors of each prime.
2. These form three identical groups: (3 × 5), (3 × 5) and (3 × 5). Every prime factor is included in those groups.
3. Each group has product 15, so 3375 = 15³. Therefore 3375 is a perfect cube and its cube root is 15.

Q7. Explain why 156 is not a perfect square and 500 is not a perfect cube, using prime factorisation. [4 marks]

1. For the square test, factorise 156 = 2 × 2 × 3 × 13. The two factors of 2 form a matching pair.
2. The factors 3 and 13 remain unpaired, so the complete factorisation cannot form two identical groups. Therefore 156 is not a perfect square.
3. For the cube test, factorise 500 = 2 × 2 × 5 × 5 × 5. The three factors of 5 form a triplet.
4. The two factors of 2 do not form a triplet. Thus 500 cannot form three identical groups and is not a perfect cube.

Q8. Use the expressions 1729 = 1³ + 12³ and 1729 = 9³ + 10³ to explain why 1729 is a taxicab number. [2 marks]

1. A taxicab number can be written as a sum of two positive cubes in two different ways.
2. The given expressions use the distinct pairs 1 and 12, and 9 and 10, establishing this property for 1729.

Key takeaways

• Squaring uses two equal factors, while cubing uses three; their geometric pictures are square arrays and cubes built from layers.
• Positive square numbers have an odd number of distinct factors, explaining which lockers remain open in the toggling puzzle.
• A forbidden units digit can rule out a perfect square, but an allowed digit alone cannot confirm one.
• The first n odd numbers, beginning with 1, add to n²; repeated subtraction uses the same relationship in reverse.
• Prime factors of perfect squares form complete pairs, while prime factors of perfect cubes form complete triplets.
• Square-root estimation places the answer between known bounds; an approximate value must not be treated as an exact equality.
• Two consecutive triangular numbers combine into a square, while successive differences reveal further patterns in the square sequence.
• The number 1729 is the smallest sum of two positive cubes expressible in two different ways.

Test yourself

Why does locker 6 finish closed in the 100-locker puzzle?

Its factors are 1, 2, 3 and 6, so it undergoes four toggles. Starting closed, an even number of toggles leaves it closed.

Which units digits rule out a perfect square immediately?

A number ending in 2, 3, 7 or 8 cannot be a perfect square.

What does reaching zero after five successive odd-number subtractions from 25 tell you?

It shows that 25 is the sum of the first five odd numbers, so 25 = 5².

How many trailing zeroes does the square of a number with three trailing zeroes have?

It has six trailing zeroes, because squaring doubles the number of zeroes at the end.

How do the two integer square roots of 64 differ from the value √64?

The two integer roots are 8 and −8. The symbol √64 denotes the positive root, 8.

Why is 9 not a perfect cube?

It lies between consecutive cubes 2³ = 8 and 3³ = 27, so it is not itself a perfect cube.

How do four layers of 16 unit cubes explain 4³?

Each layer has 4 × 4 unit cubes. Four such layers contain 4 × 4 × 4 = 64 unit cubes.

What must be checked before taking one factor from each triplet to find a cube root?

Every prime factor must belong to a complete triplet of equal factors, with none left unmatched.