Number Play | CBSE Class 6 Maths Notes

This note covers numbers as information, taller neighbours, supercells, number lines, digit sums, palindromes, the Kaprekar constant, clock and calendar patterns, mental arithmetic, number patterns, the Collatz conjecture, estimation, and winning strategies in number games.

How can numbers describe an arrangement of children?

A number can communicate something about an arrangement. In the height activity, a neighbour means a child standing immediately next to another child. Each child reports how many of their neighbours are taller than they are.

The number describes a comparison with nearby children. It does not give the child's height or count every taller child in the line. Rearranging the children can therefore change the numbers they report, even though their heights remain unchanged.

What changes at the ends of the line?

A child at either end has just one neighbour. That child cannot report 2. An interior child, meaning one with a neighbour on each side, must compare both neighbours before reporting a number.

For five children of different heights, the tallest child reports 0 wherever they stand. No neighbour can be taller. This explains why an arrangement in which all five report 1 is impossible.

Worked example 1. Five children have different heights. Can four report 1 and the last report 0?

Answer: Yes. Arrange them from shortest to tallest. Each child except the tallest has exactly one taller neighbour, on the side towards the taller end. The tallest has none. The reported sequence, meaning the numbers in order, is 1, 1, 1, 1, 0.

When testing an arrangement, check each position separately. A proposed sequence is possible only if every reported number agrees with the neighbouring heights. The end positions and the tallest child provide useful checks before trying more complicated arrangements.

How do you recognise supercells in rows and grids?

Definition: A supercell is a cell whose number is greater than the numbers in all its neighbouring cells. A cell is one box in a row or grid, and adjacent cells are the boxes immediately beside it.

In a single row, compare the left and right neighbours. An end cell has only one neighbour. In a grid with several rows, neighbours lie immediately above, below, left and right. Diagonal cells, which meet at a corner, are excluded from this comparison.

The word greater matters: equality does not satisfy the condition. Also, being greater than one neighbour is insufficient when the cell has other neighbours. Every relevant comparison must succeed.

Worked example 2. Decide whether 626, between 577 and 345, is a supercell. Also check 198 at the end of a row, beside 109.

Answer: 626 is greater than both 577 and 345, so it is a supercell. The end value 198 is greater than its single neighbour 109, so it is also a supercell.

Property: Neighbouring cells cannot both be supercells

If one cell's number is greater than the next, the next cannot also be greater than the first. To create many supercells in a row, separate higher numbers with lower numbers. With distinct numbers, the largest entry must be a supercell.

The four-row grid highlights 7500, 9870, 8632 and 6034. Around 8632 are 4795 above, 1944 below, 4580 to the left and 8280 to the right.

Reference: NCERT Class 6, page 58, Table 1

8632 exceeds each of those four neighbours. The larger value 9124 lies diagonally from it and does not disqualify it. This shows why identifying the correct neighbours must come before comparing their values.

How do number lines reveal order and spacing?

A number line places numbers along a line so that equal distances represent equal differences in value. On the number lines here, values increase towards the right. A number's position tells you both its order and its distance from other marked values.

An interval is the gap between two positions. Before filling missing labels, work out how much one equal interval represents. Two labelled positions may have an unlabelled position between them, so their difference may cover more than one interval.

How can you locate a number between thousands?

First find the two neighbouring thousand marks. Then compare the number with those values. Both 2180 and 2754 belong between 2000 and 3000. Their positions within that interval differ because 2180 is nearer 2000, while 2754 is nearer 3000.

Worked example 3. Locate 1500 and 3050 on a number line marked at 1000, 2000, 3000, 4000 and successive thousands.

Answer: 1500 lies halfway between 1000 and 2000 because it is 500 from either end. The number 3050 lies just to the right of 3000 because it is 50 more than 3000.

Line (a) labels 2010 and 2020 with one tick between them. Line (b) labels consecutive ticks 9996 and 9997. Line (c) begins with 15,077 and 15,078 on consecutive ticks.

Reference: NCERT Class 6, page 59, unnumbered number lines

Line (a) therefore increases by 5 per interval, while lines (b) and (c) increase by 1. The same physical spacing can represent different numerical differences on different lines. Establish the scale afresh rather than transferring it from another diagram.

After completing a line, read it from left to right to check the order. The smallest displayed number is at the leftmost marked position and the largest is at the rightmost marked position.

How can you investigate digits and digit sums?

A digit is a symbol used to write a number, such as 6 or 8 in 68. The digit sum is obtained by adding the individual digits. The signs + and = mean “add” and “is equal to” respectively.

Place value means the value a digit has because of its position. Units count ones; tens count groups of ten.

The numbers 68, 176 and 545 look different and have different values, but their digit sums agree: 6 + 8 = 14, 1 + 7 + 6 = 14, and 5 + 4 + 5 = 14.

Worked example 4. Find the smallest number with digit sum 14 and the largest five-digit number with digit sum 14.

Answer: The smallest is 59. A single digit cannot total 14; in two digits, the units digit can supply at most 9, leaving at least 5 for the tens digit. The largest five-digit number is 95,000: put 9 first, the remaining 5 next, and zeros afterwards.

How do digit counts differ from number counts?

Counting appearances of a digit means counting its position each time, even when one number contains that digit more than once.

Worked example 5. How often does the digit 7 appear when all numbers from 1 to 100 are written?

Answer: It appears ten times in the units position, from 7 through 97 at intervals of ten. It appears ten times in the tens position, in 70 through 79. The total is 20 appearances; 77 contributes two.

For counting numbers by length, begin with the counting numbers 1 to 9. These give nine one-digit numbers. The two-digit numbers extend from 10 to 99, giving 90 numbers; the three-digit numbers extend from 100 to 999, giving 900.

A larger number need not have a larger digit sum. Adding zeros after the digits of 95 gives larger numbers while preserving digit sum 14. Thus, without a restriction on the number of digits, there is no largest number with that digit sum.

What are palindromes and how does reverse-and-add work?

Definition: A palindrome, or palindromic number, reads identically from left to right and from right to left. Examples include 66, 848, 575, 797 and 1111.

Property: Matching positions in a palindrome have matching digits

Compare the first digit with the last, then move inwards. In a three-digit palindrome, the outer digits must agree; the middle digit can be chosen separately. Using the digits 1, 2 and 3, repetition is allowed, as in 222.

The complete list is 111, 121, 131, 212, 222, 232, 313, 323 and 333. Organising the list by its first digit helps prevent omissions and repetitions. There are three choices for the outer digit and three choices for the middle digit.

What steps should you repeat?

1. Start with a two-digit number.
2. Reverse its digits to make another number.
3. Add the original number and its reverse.
4. Stop if the sum is a palindrome. Otherwise, reverse the new sum and repeat the addition.

Worked example 6. Apply reverse-and-add to 48 until a palindrome appears.

Answer: Reversing 48 gives 84. Add them: 48 + 84 = 132. Since 132 is not a palindrome, reverse it to get 231. Now 132 + 231 = 363, which reads the same in both directions, so stop.

Further examples include 34 + 43 = 77 and 29 + 92 = 121. These reach a palindrome after one addition, whereas 48 needs another round. Do not assume that every starting number needs the same number of rounds.

Note: Starting with a two-digit number, repeated reversing and adding does give a palindrome. For three-digit numbers, the general answer is unknown. It is suspected that starting with 196 never yields a palindrome.

Suspected describes an unproved possibility. A suspicion about 196 is not a proved result, and successful trials with other numbers do not settle it.

How does Kaprekar's procedure reach 6174?

D.R. Kaprekar, a mathematics teacher in Devlali, Maharashtra, discovered the four-digit pattern in 1949. The procedure begins with a four-digit number having at least two different digits. Keep that starting condition attached to the result.

Let A mean the largest arrangement of the current digits, B the smallest arrangement, and C their difference, the amount left after subtraction. The sign − means “subtract”. Thus, C = A − B means subtract B from A.

1. Choose a permitted four-digit starting number.
2. Arrange its digits from largest to smallest to obtain A.
3. Arrange the same digits from smallest to largest to obtain B.
4. Subtract to obtain C. If C has fewer than four digits, add leading zeros to make four digit positions before the next round. Retain all four digits when forming A and B, allowing B to begin with zero.

Worked example 7. Apply the procedure to 6382.

Answer: The first round gives 8632 − 2368 = 6264. Rearranging those digits gives 6642 − 2466 = 4176. Rearranging again gives 7641 − 1467 = 6174. The number 6382 reaches 6174 in three rounds.

Property: The Kaprekar constant repeats under the procedure

A constant here is the value that stays unchanged when the same operation is repeated. From 6174, the largest arrangement is 7641 and the smallest is 1467. Their difference is 6174 again, explaining the name Kaprekar constant.

The claim is that the stated four-digit procedure reaches 6174. It is not a claim that every subtraction gives 6174, or that the original number is subtracted from its reverse. Each round requires two newly ordered arrangements of the current digits.

What patterns can you see in clocks and calendars?

A 12-hour clock displays hours in a repeating twelve-hour cycle. In a time such as 4:44, the colon separates hours from minutes. Number patterns can be recognised in the displayed digits while keeping their meaning as a time.

The examples 4:44, 10:10 and 12:21 illustrate different patterns. The first repeats a digit; the second repeats a pair; the third reads the same in both directions when the colon is ignored. Repetition alone does not make every display a palindrome.

How can you find the next palindromic time?

Worked example 8. The time is 10:01 on a 12-hour clock. Find the next palindromic time and the one after that.

Answer: The next is 11:11, after 70 minutes. The following one is 12:21, another 70 minutes later, or 140 minutes after 10:01. In each display, ignoring the colon leaves digits that read identically forwards and backwards.

Keep the starting time clear when giving an elapsed time, meaning the time that has passed. “Another 70 minutes” and “140 minutes after 10:01” describe the same later display from different starting points.

How do dates show patterns?

In the dates here, the order is day, month and year, with slashes separating these parts. Manish's birthday, 20/12/2012, repeats the digit block 2012 when the slashes are removed. Meghana's birthday, 11/02/2011, is palindromic when its slashes are removed.

Keep leading zeros, the zeros written before other digits, such as the zero in 02, when investigating a displayed date. The pattern belongs to the written form. A repeating calendar involves more than repeated date digits: every date must fall on the corresponding day of the week.

How can mental arithmetic help you test possible results?

Mental arithmetic means calculating without writing every intermediate step. Splitting a desired total into convenient parts can help. In the addition activity, the available numbers are 25,000, 400, 13,000, 1,500 and 60,000, and they may be used repeatedly.

The sign × means “multiply”. In this activity, multiplication records repeated addition compactly: 400 × 2 means two copies of 400. The numbers on the sides can be constructed by selecting suitable numbers from the middle.

Worked example 9. Make 38,800 and 3400 from the available numbers using addition.

Answer: 38,800 = 25,000 + 400 × 2 + 13,000. Also, 3400 = 1500 + 1500 + 400. The first construction uses 400 twice, while the second uses 1500 twice.

When subtraction is also permitted, the choices change. With 40,000, 7,000, 300, 1,500, 12,000 and 800 available, the example 39,800 = 40,000 − 800 + 300 + 300 combines subtraction with repeated addition.

When is a proposed digit length impossible?

A bound is a limiting value used to decide what a result can be. Four-digit numbers cannot exceed 9999. Even adding two copies of 9999 gives 19,998, so adding two four-digit numbers cannot produce a six-digit sum.

Likewise, the smallest five-digit number is 10,000. Adding two five-digit numbers gives at least 20,000, so a total of 18,500 is impossible. These arguments examine an extreme case rather than searching endlessly for an example.

Always true means every allowed case works; only sometimes true means some work and some do not; never true means none work. One successful addition cannot establish “always”. To justify “sometimes”, show both a successful case and a case that fails.

For example, 12,350 + 24,545 = 36,895 shows that two five-digit numbers can have a five-digit sum. The upper possibilities must also be considered before deciding whether that description applies to every such addition.

How can grouping make number-pattern calculations quicker?

A visual arrangement can reveal a shorter calculation. Instead of adding each entry in turn, group equal entries and count how many copies there are. A group total is the sum of the entries in one such group.

This method is useful when a design repeats the same number in rows, columns or other clearly separated regions. The shape helps organise the count, but the numerical value of each entry still matters. Equal numbers of boxes need not represent equal totals.

The pattern has three rows of four boxes labelled 40, alternating with two rows of five boxes labelled 50. The rows of 50 extend farther sideways than the rows of 40.

Reference: NCERT Class 6, page 67, pattern a

Worked example 10. Find the total in that pattern of three rows of four 40s and two rows of five 50s.

Answer: The three rows contain twelve copies of 40, giving 480. The other two rows contain ten copies of 50, giving 500. Adding the two group totals gives 480 + 500 = 980.

How can different groups have equal totals?

In the rectangular pattern labelled (c), the upper part has four rows of eight entries labelled 32. The lower part has four rows containing four entries labelled 64 in each row, with a gap separating three entries from the final entry.

The upper part therefore contains 32 copies of 32, and the lower part contains 16 copies of 64. Each part totals 1024, so together they total 2048. Doubling the value while halving the number of entries preserves the total in these two groups.

Check the layout before multiplying. In particular, do not count an empty gap as an entry or count a boundary entry twice. A clear grouping should account for every printed number exactly once.

What is the Collatz conjecture and what remains unknown?

A remainder is the amount left after forming complete equal groups. An even number can be divided by 2 without a remainder; an odd number cannot. The Collatz procedure chooses its next step according to whether the current number is even or odd.

1. Start with a positive whole number, meaning a counting number greater than zero.
2. If it is even, divide it by 2, taking half.
3. If it is odd, multiply it by 3 and add 1.
4. Apply the same test to each new number, recording the sequence until 1 is reached in the examples.

Worked example 11. Apply the Collatz rule starting with 12.

Answer: The sequence is 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. For instance, half of 12 is 6; half of 6 is 3; multiplying the odd number 3 by 3 and adding 1 gives 10. Continue testing each new number.

Why is a conjecture different from a proof?

A conjecture is a proposed mathematical claim that has not been proved. A proof is reasoning that establishes a claim for every case covered by it. Lothar Collatz proposed in 1937 that this procedure would reach 1 for every positive whole-number starting value.

The general problem remains unsolved. The displayed sequences beginning with 12, 17, 21 and 22 reach 1, but these examples do not prove the claim for all starting numbers. A successful calculation settles that particular starting value.

The sequence beginning with 21 is 21, 64, 32, 16, 8, 4, 2, 1. Once it reaches 64, repeated halving is enough. The values from 64 down to 1 are powers of 2, numbers formed by multiplying copies of 2, together with 1 at the start of the powers sequence.

This explains why the powers of 2 reach 1 under repeated halving. It does not explain why every other positive starting number must eventually reach a power of 2. That missing general argument must not be replaced with an assumption.

How do you make and explain a useful estimate?

An estimate is an approximate value sufficient for a particular purpose when an exact count is unavailable or unnecessary. Approximate means close rather than exact. A good explanation identifies the known information and the assumption used to extend it.

An assumption is something taken as true for a calculation without having counted or checked it directly. In a school estimate, one class can provide a useful starting point, but assuming similar numbers in other classes does not turn that estimate into an exact enrolment count.

Worked example 12. Paromita's three class sections contain 32, 29 and 35 children. Her school has Classes 6 to 10, each with three sections. Estimate the school total, assuming similar numbers in each class.

Answer: Her class contains 32 + 29 + 35 = 96 children, or about 100. Classes 6 to 10 give five classes. Assuming about 100 children in each class gives an estimated school total of around 500.

What should remain approximate?

The three given section counts produce an exact class total of 96. Replacing that with about 100 simplifies the estimate. The school figure also depends on the separate assumption that the other classes have similar numbers.

Words such as about and around carry information: they tell the reader not to treat the result as an exact count. Removing them changes the meaning of the answer. State the purpose of the estimate before deciding how much accuracy is needed.

Other estimation activities include steps along a route, breaths in a given time, and the capacity of containers. The relevant observations depend on the situation. A distance, cost or duration cannot be calculated exactly from an activity description that supplies no measurements or prices.

Compare an estimate with an exact count when one becomes available. The comparison helps assess whether the method was suitable and whether the assumption about similar quantities needs reconsidering.

How can a number game have a winning strategy?

A winning strategy is a plan of moves that ensures victory when followed correctly under the rules. In the game of 21, the first player says 1, 2 or 3. Players then alternate, adding 1, 2 or 3 to the previous number. Reaching 21 wins.

How can you work backwards from 21?

If you leave the running total at 17, your opponent can reach 18, 19 or 20. You can then reach 21. Work backwards in gaps of four to identify earlier totals that give the same control.

Worked example 13. Find a winning strategy for the first player in the game of 21.

Answer: Start with 1. After each opponent's addition, add the amount that makes the two additions total 4. Reply to 1 with 3, to 2 with 2, and to 3 with 1. Your totals are then 1, 5, 9, 13, 17 and 21.

Every reply is an allowed move. The plan specifies what to do for every possible opponent move, which is why it is a strategy rather than one lucky sequence of turns. The opponent's choices cannot change the combined increase of four.

How does the strategy change for 99?

In the variation, the first player says a number from 1 to 10. Each later turn adds a number from 1 to 10, and reaching 99 wins. The second player can make each pair of additions total 11.

The second player then controls the totals 11, 22, 33, 44, 55, 66, 77, 88 and 99. Each reply is 11 minus the opponent's addition, so it remains between 1 and 10. Changing the winning number or allowed additions requires reconsidering the strategy.

Developing and following precise procedures is part of computational thinking. The height comparisons, digit procedures and number games all practise this skill: identify the rule, apply it carefully, and explain why it produces the observed result.

Glossary

• Neighbour — A person or cell immediately beside the one being considered in an arrangement.
• Supercell — A cell whose number is greater than every number in its specified neighbouring cells.
• Number line — A line on which equal distances represent equal numerical differences between positions.
• Digit sum — The total obtained by adding the individual digits used to write a number.
• Place value — The value a digit represents because of its position within a written number.
• Palindrome — A number whose digits read identically from left to right and from right to left.
• Reverse-and-add — A procedure that reverses a number's digits and adds the reversed number to it.
• Kaprekar constant — The number 6174, which repeats under the stated four-digit rearrangement and subtraction procedure.
• Sequence — Numbers written in an order, such as those generated by repeatedly applying a rule.
• Conjecture — A proposed mathematical claim whose truth has not yet been established by a proof.
• Estimate — An approximate value used when an exact count is unnecessary or is not available.
• Assumption — A statement accepted for a calculation without directly checking or counting what it describes.
• Winning strategy — A plan of permitted moves that ensures victory when followed correctly in a game.
• Computational thinking — Thinking about and formulating set procedures for using numbers to carry out different purposes.

Common errors and misconceptions

• Misconception: A child's reported number counts every taller child in the line. Correct: It counts taller immediate neighbours; an end child has only one neighbour.
• Misconception: A grid cell must exceed diagonal entries to be a supercell. Correct: Compare only the immediate left, right, upper and lower neighbours.
• Misconception: Every number line increases by one at each tick. Correct: Use labelled values and the number of equal intervals to establish the scale.
• Misconception: Counting occurrences of 7 means counting each number containing it once. Correct: Count every appearance, so 77 contributes two.
• Misconception: Reverse-and-add is known to reach a palindrome for every three-digit number. Correct: The general answer is unknown; it is suspected that 196 never yields one.
• Misconception: Any four-digit starting number satisfies the stated Kaprekar condition. Correct: The starting number must have at least two different digits.
• Misconception: A few successful Collatz sequences prove the whole conjecture. Correct: They establish those cases; the general problem remains unsolved.
• Misconception: Paromita counted exactly 500 students in her school. Correct: Around 500 is an estimate based on about 100 per class and similar class sizes.

Exam-style questions with model answers

Q1. A supercell exceeds all its immediate neighbours. In a row, 626 has neighbours 577 and 345; the end value 198 has only neighbour 109. Decide whether each is a supercell. [2 marks]

1. 626 is a supercell because it is greater than both neighbouring numbers, 577 and 345.
2. 198 is also a supercell because an end cell needs to exceed only its single neighbour, here 109.

Q2. Find the digit sums of 68 and 176. What do the results show about different numbers? [2 marks]

1. The digit sum of 68 is 6 + 8 = 14.
2. The digit sum of 176 is 1 + 7 + 6 = 14. Different numbers can therefore have the same digit sum.

Q3. Starting with 48, reverse its digits and add. Repeat if necessary until a palindrome appears. Show the two additions and explain when to stop. [3 marks]

1. Reverse the starting number 48 to obtain 84. Adding gives 48 + 84 = 132, the result of the first round.
2. The number 132 does not read the same backwards. Reverse this new result to get 231, then calculate 132 + 231 = 363.
3. Stop at 363 because its digits read identically from left to right and from right to left. It is a palindrome reached after two additions.

Q4. Start with 6382. In each round, arrange the current digits into the largest and smallest numbers, then subtract the smaller from the larger. Show the three rounds reaching 6174 and explain why 6174 then repeats. [4 marks]

1. Using the digits of 6382, the largest arrangement is 8632 and the smallest is 2368. Their difference is 6264.
2. Using the digits of 6264, form 6642 and 2466. Subtracting the smaller from the larger gives 4176.
3. Using the digits of 4176, form 7641 and 1467. Their difference is 6174, reached after the third round.
4. Rearranging 6174 again gives 7641 and 1467, whose difference is still 6174. This repeating value is the Kaprekar constant.

Q5. Paromita's Class 6 sections contain 32, 29 and 35 children. The school has Classes 6 to 10, with three sections in each class. Assume other classes contain similar total numbers. Find the exact Class 6 total, estimate the school total, and explain the assumption and limits of that estimate. [5 marks]

1. Add the three supplied section counts: 32 + 29 + 35 = 96. This is the exact Class 6 total because every section count is given.
2. Use about 100 as a convenient estimate for the 96 children in that class. Keep “about” because 100 is not the exact count.
3. Classes 6, 7, 8, 9 and 10 give five classes altogether. Each has three sections, as stated in the question.
4. Assuming similar total numbers in each class, estimate about 100 children per class. Multiplying 100 by 5 gives around 500 children in the school.
5. The school figure remains approximate: the other classes were assumed to be similar, and their actual section counts were not supplied.

Q6. Apply the Collatz rule to 12: halve an even number, or multiply an odd number by 3 and add 1. Give the sequence to 1, explain the first odd-number step, and state whether this calculation proves that all positive starting numbers reach 1. [3 marks]

1. The sequence is 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. Each new value is obtained by applying the rule to the preceding value.
2. The first odd value is 3. Multiply it by 3 and add 1 to obtain 10. Then halve 10 because it is even.
3. This calculation proves that the starting value 12 reaches 1. It does not prove the claim for all positive starting numbers; the general Collatz conjecture remains unsolved.

Q7. In the game of 21, the first player says 1, 2 or 3. Players then alternate adding 1, 2 or 3 to the previous total. The first to reach 21 wins. Explain a guaranteed winning strategy for the first player. [5 marks]

1. The first player should begin by saying 1. This sets up a sequence of totals that can be maintained despite the opponent's choices.
2. After the opponent adds 1, add 3; after 2, add 2; after 3, add 1. Each reply lies within the permitted choices.
3. Every opponent move and reply together increase the total by 4. The first player's totals therefore become 1, 5, 9, 13 and 17.
4. After the first player reaches 17, the opponent can reach only 18, 19 or 20, because the largest permitted addition is 3.
5. The first player then adds 3, 2 or 1 respectively to reach 21 and win. The reply rule covers every possible opponent move.

Q8. A pattern has three rows containing four copies of 40 each, and two rows containing five copies of 50 each. Calculate the total by grouping equal entries and explain why the grouping is complete. [3 marks]

1. The three rows of 40 contain twelve entries altogether. Multiplying twelve by 40 gives a group total of 480.
2. The two rows of 50 contain ten entries altogether. Multiplying ten by 50 gives a second group total of 500.
3. Add the group totals: 480 + 500 = 980. Every entry belongs to exactly one of the two groups, so none is omitted or counted twice.

Key takeaways

• Numbers can describe relationships within arrangements, so first identify exactly what is being counted or compared.
• A supercell must exceed every specified neighbour; in a grid, diagonal cells do not enter the comparison.
• Read the scale of a number line before filling gaps, and distinguish digit sums from the values of numbers.
• Palindromes read identically in both directions; repeated reversing and adding can require more than one round.
• Kaprekar's four-digit procedure starts with at least two different digits and reaches the repeating value 6174.
• Grouping equal entries simplifies addition, while extreme possible values help explain why some requested results are impossible.
• Successful Collatz examples do not prove the general conjecture, which remains an unsolved mathematical problem.
• Estimates need stated assumptions, and winning strategies need a response to every permitted move by an opponent.

Test yourself

Why can an end child not report two taller neighbours?

An end child has only one immediate neighbour, so cannot have two taller immediate neighbours.

In a grid, 8632 has immediate neighbours 4795, 1944, 4580 and 8280. Is it a supercell?

Yes. The number 8632 is greater than every one of its four immediate neighbours.

A number line has two equal intervals from 2010 to 2020. What does each interval represent?

The total increase is 10 across two intervals, so each interval represents an increase of 5.

What is the digit sum of 545?

It is 14, obtained by adding the digits: 5 + 4 + 5 = 14.

Why does 6174 repeat under Kaprekar's procedure?

Its largest digit arrangement is 7641 and its smallest is 1467; subtracting gives 6174 again.

Under the Collatz rule, what follows the odd number 21?

Multiply 21 by 3 and add 1 to obtain the next number, 64.

Why is Paromita's school total of around 500 an estimate?

It uses about 100 children per class and assumes the other classes have similar numbers.

In the game of 21, after starting at 1, how should you reply when your opponent adds 2?

Add 2 yourself, making the two additions total 4 and preserving the planned sequence of totals.