Number line, comparing numbers and supercells
In a queue, the tallest child stands out because everyone near her is shorter. Numbers behave the same way: some numbers are bigger than all their neighbours. On a ruler, numbers on the right are always bigger.
On a number line, a number to the right is greater than a number to its left. To compare numbers, first count the digits (more digits means bigger), then compare from the left, place by place. A supercell is a cell in a grid whose number is greater than the numbers in all neighbouring cells that share a side with it.
In the figure: A and B are at 3 and 8. The middle point is (3 + 8) ÷ 2 = 5.5.
Rule 1: Compare and find the middle — More digits means a bigger number. With the same number of digits, compare from the left: the first place where the digits differ decides. The number exactly in the middle of two numbers is their sum divided by 2.
Number line, comparing numbers and supercells
In the app you can move this figure with your finger.
Open in the app- Easy: 4,827 and 4,872: thousands and hundreds are equal; tens 2 < 7, so 4,872 is bigger.
- Medium: Middle of 120 and 180: (120 + 180) ÷ 2 = 150.
- Exam-type: Whole numbers strictly between 4,999 and 5,010 are 5,000 to 5,009, which is 10 numbers.
Rule 2: Find the supercells — Check each cell against its neighbours above, below, left and right. If it beats every one of them it is a supercell. Two neighbouring cells can never both be supercells.
Number line, comparing numbers and supercells
In the app you can move this figure with your finger.
Open in the app- Easy: Row 1: 7, 2; Row 2: 5, 9. The cell 7 beats 2 and 5; the cell 9 beats 2 and 5. Supercells: 2.
- Medium: List 4, 8, 3, 9, 2: the inner numbers 8 and 9 beat both neighbours. That is 2.
- Exam-type: Five numbers in a row can have at most 3 supercells (cells 1, 3 and 5), because supercells cannot touch.
The greatest number in a grid is always a supercell. Neighbours share a side, not just a corner.
Comparing only the last digits of two numbers, or counting corner cells as neighbours.
Count digits first; then compare from the left. Middle = (a + b) ÷ 2. Memory clue: “Longer wins, then Left to right”. For supercells: “Beats all four sides”.
Spot the difference
| Case | Rule | Example | Answer |
|---|---|---|---|
| Different digits | more digits is bigger | 56 and 506 | 506 |
| Same digits | first different place decides | 4,827 and 4,872 | 4,872 |
| Middle number | (a + b) ÷ 2 | 250 and 450 | 350 |
| Supercell | beats all side neighbours | 9 among 1, 3, 2, 4 | yes |
Practice questions
- 1Which is greater, 4,827 or 4,872?
- 2Which number comes just after 9,999?
- 3What is exactly in the middle of 10 and 20?
- 4Greatest 4-digit number from the digits 2, 0, 7, 5?
- 5Smallest 4-digit number from the digits 2, 0, 7, 5?
- 6Halfway between 250 and 450?
- 7Grid: Row 1: 4, 8, 1; Row 2: 6, 2, 9; Row 3: 3, 7, 5. How many supercells?
- 8How many whole numbers lie strictly between 4,999 and 5,010?
Show answers (try them yourself first)
- 11) 4,872
- 22) 10,000
- 33) 15
- 44) 7,520
- 55) 2,057 (no zero in front)
- 66) 350
- 77) 4 (8, 6, 9 and 7)
- 88) 10 (5,000 to 5,009)
Digits and digit sums
A bus ticket number 3408 can be broken into the digits 3, 4, 0 and 8. A detective checks each digit, adds them and finds clues. That is how “digit detectives” work with numbers.
The digits are 0 to 9. The digit sum of a number is the sum of all its digits. The place value of a digit is the digit times its place: in 3,742 the 7 stands for 700. A 4-digit number cannot start with 0.
In the figure: The digits of 3,408 are 3, 4, 0 and 8. Their sum is 15.
Rule 1: Add the digits — Add all the digits. Keep adding the digits of the answer until one digit is left; this single digit is the digit sum in its simplest form.
Digits and digit sums
In the app you can move this figure with your finger.
Open in the app- Easy: Digit sum of 345: 3 + 4 + 5 = 12.
- Medium: 4,567: 4 + 5 + 6 + 7 = 22, then 2 + 2 = 4.
- Exam-type: Which has the greatest digit sum: 399 (21), 480 (12), 570 (12) or 888 (24)? 888.
Rule 2: Make numbers with a given digit sum — For the smallest number, keep the left digits as small as possible and push the big digits to the right. For the largest number, put as much as possible in the leftmost place.
- Easy: Smallest 2-digit number with digit sum 9: tens 1, ones 8, so 18.
- Medium: Largest 3-digit number with digit sum 5: 500. Smallest: 104.
- Exam-type: Smallest number with digit sum 20: two digits give at most 18, so use 3 digits: 299.
The digit sum of 99 is 18, the biggest for 2 digits. A single-digit number is its own digit sum.
Taking the first digit as 0 for a 3- or 4-digit number, for example calling 057 a 3-digit number.
Smallest number: small digits on the left. Largest number: big digits on the left. Memory clue: “Small Left for Small, Big Left for Big”.
Spot the difference
| Digit sum | Smallest number | Largest 3-digit number | Count of 2-digit numbers |
|---|---|---|---|
| 4 | 4 | 400 | 4 |
| 9 | 9 | 900 | 9 |
| 15 | 69 | 960 | 4 |
| 20 | 299 | 992 | 0 |
Practice questions
- 1Digit sum of 345?
- 2Digit sum of 1,000?
- 3Place value of 7 in 3,742?
- 4Smallest 3-digit number with digit sum 5?
- 5How many 2-digit numbers have digit sum 4?
- 6Largest 4-digit number with digit sum 6?
- 7How many 3-digit numbers have digit sum 3?
- 8Smallest number with digit sum 20?
Show answers (try them yourself first)
- 11) 12
- 22) 1
- 33) 700
- 44) 104
- 55) 4 (13, 22, 31, 40)
- 66) 6,000
- 77) 6 (102, 111, 120, 201, 210, 300)
- 88) 299
Palindromes and reverse-and-add
Words like MADAM and RADAR read the same from both ends. Numbers can do that too: 121 and 4,554 stay the same when written backwards.
A palindrome reads the same forwards and backwards. The reverse of a number writes its digits in the opposite order. In reverse-and-add, add a number and its reverse, then repeat with the answer until you get a palindrome.
In the figure: 68 + 86 = 154, 154 + 451 = 605, 605 + 506 = 1,111. Three steps give a palindrome.
Rule 1: Spot and build palindromes — A 3-digit palindrome looks like aba and a 4-digit one like abba. For 3 digits the first digit can be 1 to 9 and the middle digit 0 to 9, so there are 9 × 10 = 90 of them.
- Easy: Which reads the same both ways: 1,221, 1,231 or 1,213? 1,221.
- Medium: Next palindrome after 2,992: it must start with 3, so 3,003.
- Exam-type: Smallest 4-digit palindrome with digit sum 14: 2a + 2b = 14, a + b = 7, smallest a = 1, b = 6: 1,661.
Rule 2: Reverse-and-add steps — Reverse the number, add, and check the answer. Stop when you get a palindrome and count the steps. Most numbers need only a few steps, but some need many.
Palindromes and reverse-and-add
In the app you can move this figure with your finger.
Open in the app- Easy: 29 + 92 = 121: one step.
- Medium: 39 + 93 = 132, 132 + 231 = 363: two steps.
- Exam-type: 87: 165, 726, 1,353, 4,884. Four steps.
Every 2-digit palindrome is 11 times a digit: 11, 22, 33 … 99.
Reversing only some of the digits, or forgetting to add the original number.
Palindrome: first digit = last digit and second = second-last. Add the number to its reverse each step. Memory clue: “Mirror in the middle”: the left half and the right half match.
Spot the difference
| Start | Step 1 | Step 2 | Steps needed |
|---|---|---|---|
| 29 | 121 | already a palindrome | 1 |
| 39 | 132 | 363 | 2 |
| 57 | 132 | 363 | 2 |
| 68 | 154 | 605 | 3 |
Practice questions
- 1Reverse of 3,167?
- 2Smallest 2-digit palindrome?
- 324 + its reverse?
- 4How many 3-digit palindromes are there?
- 5Smallest palindrome greater than 1,300?
- 6Steps for 68?
- 7Steps for 87?
- 8How many 4-digit palindromes lie between 1,000 and 2,000?
Show answers (try them yourself first)
- 11) 7,613
- 22) 11
- 33) 66
- 44) 90
- 55) 1,331
- 66) 3
- 77) 4
- 88) 10 (1001, 1111, …, 1991)
Kaprekar’s constant and number patterns
A digital clock shows 12:21, a palindrome time, and a magic trick shows that almost every 4-digit number ends up at 6174 after a few easy steps. Numbers are full of surprises.
Kaprekar’s routine: take a 4-digit number with at least two different digits, arrange its digits to make the biggest and the smallest number, subtract, and repeat. You always reach 6174, called Kaprekar’s constant, and then stay there. A number pattern is a list that follows a rule such as add, double, square or triangular.
In the figure: 3,524 → 5,432 − 2,345 = 3,087 → 8,730 − 0,378 = 8,352 → 8,532 − 2,358 = 6,174.
Rule 1: Run the Kaprekar steps — Write the digits from biggest to smallest, then from smallest to biggest (zeros may come first), and subtract. Repeat with the answer written as 4 digits. A number with all four digits the same, like 2,222, cannot start the routine because the answer is 0.
Kaprekar’s constant and number patterns
In the app you can move this figure with your finger.
Open in the app- Easy: Digits 3, 1, 4, 2: 4,321 − 1,234 = 3,087.
- Medium: From 3,087: 8,730 − 0,378 = 8,352. Then 8,532 − 2,358 = 6,174.
- Exam-type: From 1,000: 999 → 8,991 → 8,082 → 8,532 → 6,174. That is 5 steps.
Rule 2: Find the pattern and clock palindromes — Look at the gaps or ratios between neighbours. Equal gaps mean add; each term twice the last means double; 1, 4, 9, 16 are squares; gaps 2, 3, 4, 5 give triangular numbers. On a 24-hour clock, times like 10:01 and 23:32 are palindromes.
- Easy: 1, 3, 6, 10, 15: the gaps are 2, 3, 4, 5, so the next gap is 6 and the next number is 21.
- Medium: 1 + 2 + … + 10: pair (1 + 10), (2 + 9) … five pairs of 11, so 5 × 11 = 55.
- Exam-type: The latest palindrome time on a 24-hour clock is 23:32. In a whole day there are 16 such times.
6174 turns into itself: 7,641 − 1,467 = 6,174.
Dropping the zeros: always write the answer with 4 digits, for example 999 as 0999, before arranging.
Big arrangement − small arrangement, repeat until 6174. Memory clue: “Big minus Small, Four digits, Stop at 6174”.
Spot the difference
| Idea | Start | Rule | Next numbers |
|---|---|---|---|
| Kaprekar | 3,524 | big − small | 3,087, 8,352, 6,174 |
| Squares | 1, 4, 9, 16 | n × n | 25, 36 |
| Doubling | 3, 6, 12 | × 2 | 24, 48 |
| Triangular | 1, 3, 6, 10 | add 2, 3, 4, 5 | 15, 21 |
Practice questions
- 1Kaprekar’s constant for 4-digit numbers?
- 2Largest number from the digits 3, 1, 4, 2?
- 3Next in 11, 22, 33, 44?
- 4One Kaprekar step on 3,087?
- 5Next in 1, 4, 9, 16, 25?
- 6Next in 1, 3, 6, 10, 15?
- 7How many Kaprekar steps from 1,000 to 6,174?
- 8Latest palindrome time on a 24-hour clock?
Show answers (try them yourself first)
- 11) 6,174
- 22) 4,321
- 33) 55
- 44) 8,352
- 55) 36
- 66) 21
- 77) 5
- 88) 23:32
Collatz sequences, mental maths, estimation and games
A ball that bounces up and down and finally settles is like a Collatz number that goes up and down before it lands on 1. Shopkeepers also add quickly in their heads and estimate bills before they use a calculator.
The Collatz rule: if the number is even, halve it; if it is odd, multiply by 3 and add 1. Keep going and every number tried so far ends at 1. Estimation means rounding numbers to get a quick, close answer. In the race to 21, players add 1, 2 or 3 in turn and whoever says 21 wins.
In the figure: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Eight steps. Odd numbers go ×3 + 1, even numbers are halved.
Rule 1: Follow the Collatz rule — Even: divide by 2. Odd: multiply by 3 and add 1. Count how many steps it takes to reach 1.
Collatz sequences, mental maths, estimation and games
In the app you can move this figure with your finger.
Open in the app- Easy: Step on 6: even, so 6 ÷ 2 = 3. Step on 7: odd, so 7 × 3 + 1 = 22.
- Medium: 5 → 16 → 8 → 4 → 2 → 1: 5 steps. 8 → 4 → 2 → 1: 3 steps.
- Exam-type: 9 climbs to 52 and takes 19 steps. 16 takes only 4 steps.
Rule 2: Mental maths and estimation — Use friendly numbers. 99 + 47 = 100 + 47 − 1. 15 × 12 = 15 × 10 + 15 × 2. 25 × 4 is four quarters of 100. To estimate, round each number first, then calculate.
- Easy: Estimate 498 + 305: 500 + 300 = 800.
- Medium: 48 + 37: 48 + 2 = 50, 37 − 2 = 35, so 85.
- Exam-type: Estimate 29 × 31: 30 × 30 = 900. Round 4,763 to the nearest thousand: 5,000.
Rule 3: Win the race to 21 — The safe numbers are 21, 17, 13, 9, 5 and 1 (4 apart). If you reach a safe number, whatever your opponent adds (1, 2 or 3), you add enough to make the total 4 more and reach the next safe number.
Collatz sequences, mental maths, estimation and games
In the app you can move this figure with your finger.
Open in the app- Easy: The safe number just before 21 is 17: from 17 the opponent can say only 18, 19 or 20.
- Medium: If the opponent says 2 after your 5, you say 2 more to reach 9.
- Exam-type: Going first, say 1. Then each round the two players together add 4.
Every Collatz number tried so far reaches 1, but nobody has proved that it always happens.
Using ×3 + 1 on an even number or halving an odd number. First check whether the number is even or odd.
Even: halve. Odd: triple and add 1. Race to 21: reach 1, 5, 9, 13, 17, 21. Memory clue: “Even Halves, Odd Triples plus One”; for the race: “Four apart wins”.
Spot the difference
| Start | Steps to reach 1 | Highest number reached | Note |
|---|---|---|---|
| 6 | 8 | 16 | even start |
| 7 | 16 | 52 | odd start |
| 9 | 19 | 52 | long path |
| 16 | 4 | 16 | shortest of these |
Practice questions
- 1Collatz step on 6?
- 2Collatz step on 7?
- 3Estimate 498 + 305.
- 4Collatz steps from 5 to 1?
- 515 × 12 in your head?
- 6Safe number just before 21 in the race?
- 7Collatz steps from 9 to 1?
- 8You go first in the race to 21. Which number do you say first, and why?
Show answers (try them yourself first)
- 11) 3
- 22) 22
- 33) 800
- 44) 5
- 55) 180
- 66) 17
- 77) 19
- 88) 1, because the safe numbers 1, 5, 9, 13, 17, 21 are 4 apart and you can always reach the next one.