1. Chapter at a Glance
Numbers are not only for counting and calculating — they can also be played with. When we play with numbers, we notice surprising facts, invent puzzles and sharpen our reasoning. This chapter is a collection of such number games: numbers that "tell" us things, supercells, digit sums, palindromes, the mysterious number 6174, the Collatz sequence, clock and calendar patterns, and quick estimation.
- Using numbers to describe arrangements (who is taller than their neighbours).
- Supercells: numbers larger than all their neighbours in a table.
- Patterns on the number line and in digit sums.
- Palindromes and the reverse-and-add game.
- The Kaprekar routine that always reaches 6174.
- The Collatz conjecture — an unsolved problem anyone can try.
- Estimation and mental-maths strategies with large numbers.
2. Numbers Can Tell Us Things
Imagine children standing in a row. Each child calls out a number: how many of their immediate neighbours are taller than them. A child at the end has only one neighbour, so can say only 0 or 1. A child in the middle has two neighbours, so can say 0, 1 or 2.
- A child who says 0 is taller than all neighbours.
- A child who says 2 is shorter than both neighbours.
- Can every child say 1? Yes — if heights increase steadily from one end to the other, then each child (except the tallest at the end) has exactly one taller neighbour. The tallest child says 0.
- Can every child say 0? No — two neighbours cannot both be taller than each other.
Such puzzles train us to reason carefully: we check every possibility and explain why something can or cannot happen.
Five children stand in a row and say 1, 0, 2, 0, 1. Is this possible? Describe a height order.
Solution: The children saying 0 (2nd and 4th) must be taller than their neighbours; the one saying 2 (3rd) must be shorter than both. Heights like 140, 155, 130, 150, 145 (in cm) work: child 1 (140) has taller neighbour 155 → says 1; child 2 (155) taller than both → 0; child 3 (130) shorter than both → 2; child 4 (150) taller than 130 and 145 → 0; child 5 (145) has taller neighbour 150 → 1. ✔
3. Supercells
In a row (or grid) of numbers, a supercell is a cell whose number is larger than all its adjacent cells. Cells at the ends have fewer neighbours, so it is easier for them to be supercells.
| Example row (supercells shown in bold) | ||||||
|---|---|---|---|---|---|---|
| 200 | 136 | 402 | 39 | 160 | 715 | 301 |
Here 200 (bigger than 136), 402 (bigger than 136 and 39) and 715 (bigger than 160 and 301) are supercells.
- Two supercells can never be next to each other — each would have to be bigger than the other.
- In a row of n cells, at most about half the cells (rounded up) can be supercells.
- The largest number in the row is always a supercell.
Fill a row of 5 cells with the digits 1–5 (each once) so that there are as many supercells as possible.
Solution: Put big numbers in alternate cells: 4, 1, 5, 2, 3. Supercells are 4, 5 and 3 — three supercells, which is the maximum for 5 cells.
4. Patterns on the Number Line and with Digits
Digit sums
The digit sum of a number is what we get by adding its digits. For example, digit sum of 368 = 3 + 6 + 8 = 17.
- Many different numbers can have the same digit sum: 14, 23, 32, 41 and 50 all have digit sum 5.
- Numbers that are multiples of 3 have digit sums that are multiples of 3 (for example, 132 → 6). Multiples of 9 have digit sums that are multiples of 9 (for example, 729 → 18).
- When you add 9 to a number (without carrying past 9 in the ones place causing chains), the digit sum often stays the same — try 23 → 32 → 41.
Counting digits
How many times does the digit 7 appear when we write all numbers from 1 to 100? In the ones place: 7, 17, 27, …, 97 → 10 times. In the tens place: 70, 71, …, 79 → 10 times. Total = 20. (The number 77 is counted twice because it has two 7s.)
Which number between 1 and 100 has the largest digit sum?
Solution: 99 has digit sum 9 + 9 = 18. No other number from 1 to 100 reaches 18.
5. Palindromes and Reverse-and-Add
A palindrome is a number that reads the same forwards and backwards, such as 121, 4554 or 70107. Words can be palindromes too: "madam", "level".
Reverse-and-add game
- Take a number, say 48.
- Reverse its digits: 84.
- Add: 48 + 84 = 132. Not a palindrome, so repeat.
- 132 + 231 = 363 — a palindrome! We stop.
Most numbers become palindromes after a few rounds. Some take many rounds (89 needs 24 rounds). For a few numbers like 196, nobody has yet found a palindrome even after millions of rounds — this is still an open question.
Palindromes with digits 1, 2, 3
Using each of 1, 2, 3 at most as needed, 3-digit palindromes have the form ABA: 111, 121, 131, 212, 222, 232, 313, 323, 333. There are 3 choices for A and 3 for B, so 9 such palindromes.
6. The Magic Number 6174 (Kaprekar Routine)
The Indian mathematician D. R. Kaprekar (1905–1986), a school teacher from Devlali in Maharashtra, discovered this amazing routine:
- Take any 4-digit number whose digits are not all the same (like 3524).
- Arrange the digits in descending order: 5432.
- Arrange them in ascending order: 2345.
- Subtract the smaller from the larger: 5432 − 2345 = 3087.
- Repeat with the answer.
Once you reach 6174, you stay there forever. It is reached in at most 7 steps from any starting number. 6174 is called Kaprekar's constant. For 3-digit numbers the routine always reaches 495.
If a result has fewer than 4 digits (like 999), treat it as 0999 and keep going. Otherwise the routine breaks.
7. Clock and Calendar Numbers
A 12-hour digital clock shows times like 1:21, 4:44, 12:21. Times that read the same both ways are palindromic times.
- From 1:01 to 9:59 the palindromic times have the form H:MH — e.g. 1:01, 1:11, 1:21, …, 1:51. Each hour has 6 such times, so 9 hours give 54.
- For 10, 11 and 12 o'clock, the form is HH:HH reversed: 10:01, 11:11, 12:21 — 3 more.
- Total palindromic times on a 12-hour clock: 57 (from 1:00 to 12:59).
Dates can be palindromes too: 12/02/2021 written as 12022021 reads the same backwards.
8. The Collatz Conjecture
Start with any counting number and follow two simple rules:
Example starting at 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Starting at 7: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
Every number tried so far — even astronomically large ones checked by computers — eventually reaches 1. But no one has proved that this will always happen. This guess, made by Lothar Collatz in 1937, is called the Collatz conjecture. It shows that even simple-looking questions in mathematics can remain unsolved.
9. Simple Estimation and Mental Maths
Estimation means finding a quick, reasonable approximate answer. It helps us check calculations and make decisions when exact values are not needed.
- How many hairs on a head? About 1 lakh — counting all is impossible, but we can count hairs in a small patch and multiply.
- Steps to walk 1 km? If one step is about 50 cm, then 1 km = 1,00,000 cm needs about 2000 steps.
- Rounding: 4,987 + 3,012 ≈ 5,000 + 3,000 = 8,000.
Mental maths strategies
| Strategy | Example |
|---|---|
| Make round numbers | 47 + 38 = 47 + 40 − 2 = 85 |
| Split into parts | 25 × 16 = 25 × 4 × 4 = 100 × 4 = 400 |
| Compensate | 396 + 205 = 400 + 201 = 601 |
| Use pairs that make 10 or 100 | 37 + 25 + 63 = (37 + 63) + 25 = 125 |
Estimate how many days you have lived if you are 11 years old.
Solution: 11 years × 365 days ≈ 11 × 365 = 4,015 days, plus about 2–3 leap days ≈ 4,018 days, i.e. roughly 4,000 days.
10. Master Formula Table
| Idea | Rule | Example |
|---|---|---|
| Digit sum | Add all digits | 4096 → 19 |
| Divisible by 3 | Digit sum divisible by 3 | 261 → 9 |
| Divisible by 9 | Digit sum divisible by 9 | 738 → 18 |
| Palindrome | Same forwards and backwards | 3443 |
| Reverse-and-add | n + reverse(n), repeat | 57 → 132 → 363 |
| Kaprekar (4 digits) | Descending − ascending → 6174 | 7641 − 1467 = 6174 |
| Kaprekar (3 digits) | Reaches 495 | 954 − 459 = 495 |
| Collatz step | Even: n ÷ 2; odd: 3n + 1 | 5 → 16 → 8 |
| 3-digit palindromes ABA | 9 × 10 = 90 | 101 to 999 |
| Supercells in a row of n | Never two side by side | At most 3 in 5 cells |
11. Master Summary Table
| Game / idea | What happens | What it teaches |
|---|---|---|
| Taller-neighbour count | Numbers describe arrangements | Reasoning about possibilities |
| Supercells | Cell bigger than its neighbours | Comparing and arranging |
| Digit sums | Patterns, divisibility | Quick checking |
| Palindromes | Same both ways | Symmetry in numbers |
| Kaprekar routine | Always reaches 6174 | Surprising fixed point |
| Collatz | Seems to always reach 1 | Open problems exist |
| Estimation | Reasonable approximate answers | Number sense |
12. Mind Map
- Taller-neighbour count
- Supercells
- Digit sums
- Counting digits
- Palindromes
- Kaprekar 6174
- Collatz → 1
- Reverse-and-add
- Clock and calendar
- Estimation
- Mental maths
13. Tips and Tricks
- For supercell puzzles, place the largest numbers in alternate positions starting from an end.
- Use digit sums to check additions: digit-sum of the answer should match the digit-sum of the parts (after reducing to one digit).
- In the Kaprekar routine, always write the number with 4 digits, adding leading zeros.
- For estimation, round to the nearest 10, 100 or 1000 first, then calculate.
- Saying two supercells can be neighbours — impossible.
- Counting 77 only once when counting the digit 7 from 1 to 100.
- Starting Kaprekar with 1111 or 2222 — all digits same gives 0.
- Thinking the Collatz conjecture is proved — it is still open.
14. Important One-liners
- A supercell holds a number larger than all its adjacent cells.
- Two supercells cannot be adjacent.
- The largest number in a row is always a supercell.
- The digit sum of 99 is 18, the largest for numbers up to 100.
- The digit 7 appears 20 times from 1 to 100.
- A palindrome reads the same forwards and backwards.
- There are 90 three-digit palindromes (101 to 999).
- Kaprekar's constant for 4-digit numbers is 6174.
- For 3-digit numbers, the Kaprekar routine reaches 495.
- D. R. Kaprekar was a mathematics teacher from Devlali, Maharashtra.
- Collatz rule: even → halve, odd → 3n + 1.
- The Collatz conjecture has not been proved.
- A 12-hour clock shows 57 palindromic times.
- Estimation gives a quick reasonable answer, not an exact one.
15. Exam-pattern Questions with Detailed Answers
Which of these numbers is a palindrome?
- 1231
- 4554
- 6789
- 1010
Show answer
4554 reversed is 4554, the same number. The others change when reversed (1231 → 1321, 6789 → 9876, 1010 → 0101).
Answer: (B) 4554
Find the digit sum of 70,825 and say whether the number is divisible by 3.
Show answer
Digit sum = 7 + 0 + 8 + 2 + 5 = 22. 22 is not a multiple of 3, so 70,825 is not divisible by 3.
Answer: 22; not divisible by 3
Write the Collatz sequence starting from 12.
Show answer
12 (even) → 6 → 3 (odd) → 10 → 5 → 16 → 8 → 4 → 2 → 1.
Answer: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1
Mark the supercells in the row: 58, 32, 77, 77, 41, 90, 12.
Show answer
58 > 32 → supercell. 77 (3rd) is not bigger than its neighbour 77, so not a supercell; the same for the 4th cell. 90 > 41 and 90 > 12 → supercell. 12 is not bigger than 90.
Answer: 58 and 90
Apply the Kaprekar routine to 2111 until you reach 6174. How many steps are needed?
Show answer
2111 − 1112 = 0999 → 9990 − 0999 = 8991 → 9981 − 1899 = 8082 → 8820 − 0288 = 8532 → 8532 − 2358 = 6174.
That is 5 steps.
Answer: 5 steps
Use reverse-and-add on 75 until you get a palindrome.
Show answer
75 + 57 = 132; 132 + 231 = 363, which is a palindrome.
Answer: 363 (after 2 rounds)
Estimate 4,896 + 2,103 + 1,995 by rounding to the nearest hundred.
- 8,800
- 9,000
- 9,100
- 8,000
Show answer
4,896 ≈ 4,900; 2,103 ≈ 2,100; 1,995 ≈ 2,000. Sum ≈ 4,900 + 2,100 + 2,000 = 9,000. (Exact value 8,994.)
Answer: (B) 9,000
How many times does the digit 5 appear when writing all numbers from 1 to 60?
Show answer
Ones place: 5, 15, 25, 35, 45, 55 → 6 times. Tens place: 50 to 59 → 10 times. Total 16 (55 has two 5s and is counted in both).
Answer: 16
Seven children stand in a row. Is it possible that the children call out 0, 2, 0, 2, 0, 2, 0? Explain.
Show answer
"0" means taller than all neighbours; "2" means shorter than both neighbours. The pattern alternates tall–short–tall–short, which is possible: heights 150, 130, 152, 128, 149, 135, 151 work.
Each 0-child is taller than its short neighbours, and each 2-child is shorter than its tall neighbours.
Answer: Yes, with tall and short children alternating
Apply the 3-digit Kaprekar routine to 321 and show that it reaches 495.
Show answer
321 − 123 = 198 → 981 − 189 = 792 → 972 − 279 = 693 → 963 − 369 = 594 → 954 − 459 = 495 → 954 − 459 = 495 (stays).
Answer: Reaches 495 in 5 steps
In a 2 × 3 grid, each cell touches the cells directly left, right, above and below it. Place the numbers 1 to 6 to get the maximum number of supercells. How many can there be?
Show answer
Supercells cannot touch each other. In a 2 × 3 grid, at most 3 cells can be chosen with none touching — in a checkerboard pattern (top-left, top-right, bottom-middle).
Place 6, 5, 4 in those cells and 1, 2, 3 in the others, e.g. top row 6, 1, 5 and bottom row 2, 4, 3. Supercells: 6, 5, 4.
Answer: 3 supercells
Estimate the number of words in a 200-page book if a typical page has 30 lines with about 11 words per line.
Show answer
Words per page ≈ 30 × 11 = 330 ≈ 300 (round down a little for half-empty lines).
Words in the book ≈ 200 × 330 = 66,000, so roughly 60,000–66,000 words.
Answer: About 66,000 words
16. Quick Revision Summary
- Numbers can describe arrangements; reasoning tells us which are possible.
- Supercells are bigger than all neighbours and are never adjacent.
- Digit sums reveal divisibility by 3 and 9.
- Palindromes read the same both ways; reverse-and-add usually creates one.
- Kaprekar: 4-digit numbers → 6174; 3-digit numbers → 495.
- Collatz: halve evens, 3n + 1 for odds — always seems to reach 1, but unproved.
- Estimation and mental maths make calculations faster and help check answers.