Mental Ability · Lesson 8

Data Handling & Number Puzzles

Turn a messy pile of facts into tallies, pictographs and bar graphs — and read them like a detective. Then crack number patterns, magic squares and secret-digit puzzles with smart shortcuts.

~40 min read7 quick checksTest: 20 questions

1Collecting and organising data

Data is a collection of facts or numbers — like the favourite colours of your classmates or the runs scored in each over. Data that is just listed as it was collected is called raw data. It is hard to read until we organise it.

✏️ Worked example — from raw data to a table

Twelve children were asked their favourite fruit: mango, apple, mango, banana, grapes, mango, apple, banana, mango, grapes, mango, apple.

  1. List each different fruit once: mango, apple, banana, grapes.
  2. Go through the list and make a mark for each answer (a tally).
  3. Count the marks to get the frequency (how many times each appears): mango 5, apple 3, banana 2, grapes 2.
  4. Check: 5 + 3 + 2 + 2 = 12 = number of children asked ✓
FruitTally marksFrequency
Mango卌5
Apple|||3
Banana||2
Grapes||2
Total12

Tally marks

Draw one line for each item. The fifth line is drawn across the first four to close a "gate" of 5. Then count in fives — fast and hard to get wrong.

55 + 5 + 3 = 13
Each gate = 5. Two gates and 3 single lines = 5 + 5 + 3 = 13.
⚠️ Olympiad trap — counting the crossing line

A gate has 4 standing lines + 1 crossing line = 5, not 4 and not 6. Count the gates first (× 5), then add the single lines.

How would you show 23 with tally marks?

  1. 2 gates and 3 lines
  2. 3 gates and 8 lines
  3. 4 gates and 3 lines
  4. 5 gates and 3 lines
4 gates = 4 × 5 = 20, plus 3 lines = 23. (Never leave 5 or more single lines — close them into a gate.)

2Pictographs

A pictograph (picture graph) shows data using pictures or symbols. Every pictograph has a key (also called the scale) that tells you how much one symbol stands for. Always read the key first!

Trees planted by each class Class 4A Class 4B Class 4C Class 4D Key: each tree = 10 trees planted
A half symbol stands for half of the key: ½ × 10 = 5 trees.
✏️ Worked example — reading the pictograph
  1. Class 4A: 4 symbols × 10 = 40 trees.
  2. Class 4B: 3½ symbols → 3 × 10 + 5 = 35 trees.
  3. Class 4C: 5 × 10 = 50 (the most). Class 4D: 2½ → 25 (the fewest).
  4. Total = 40 + 35 + 50 + 25 = 150 trees. (Shortcut: count symbols first: 4 + 3½ + 5 + 2½ = 15 symbols × 10 = 150.)
  5. How many more must 4D plant to equal 4C? 50 − 25 = 25 (that's 2½ symbols).
✏️ Worked example — drawing a pictograph

Key: one ☺ = 6 children. How many symbols show 45 children?

  1. 45 ÷ 6 = 7 remainder 3.
  2. 3 is half of 6, so the remainder is ½ symbol.
  3. Answer: 7½ symbols.
⚠️ Olympiad trap — counting symbols instead of values

If the question asks "how many trees", answer in trees (symbols × key), not in symbols. And check the key every time — a different graph may use a key of 4, 5 or 100.

In a pictograph one 🚲 stands for 8 bicycles. How many 🚲 symbols are needed to show 36 bicycles?

  1. 4
  2. 5
  3. 4½
  4. 3½
36 ÷ 8 = 4 remainder 4. 4 is half of 8, so it is 4½ symbols.

3Bar graphs

A bar graph shows data with bars of equal width. The length of each bar shows the value. One axis names the items; the other axis has a scale — equally spaced numbers.

Ice-cream cups sold on Sunday 0 10 20 30 40 50 60 70 80 Vanilla Chocolate Mango Kesar Pista Number of cups Scale: 1 small line gap = 10 cups
A bar ending halfway between 40 and 50 shows 45.
✏️ Worked example — reading the bar graph
  1. Read each bar: Vanilla 45, Chocolate 75, Mango 55, Kesar 25, Pista 40.
  2. Most popular: Chocolate (tallest bar); least popular: Kesar.
  3. How many more Chocolate than Kesar? 75 − 25 = 50.
  4. Total cups sold: 45 + 75 + 55 + 25 + 40 = 240.
  5. Which two flavours together sold as many as Chocolate and Kesar together (75 + 25 = 100)? Vanilla + Mango = 45 + 55 = 100 ✓
  6. What fraction of all cups were Chocolate or Vanilla? (75 + 45) = 120 out of 240 = ½.

Drawing a bar graph

  1. Draw two axes. Write the items along one axis and give the graph a title.
  2. Choose a sensible scale — e.g. 1 unit = 10 cups if values go up to 75. The scale must go up in equal steps and start from 0.
  3. Draw bars of the same width with equal gaps between them.
  4. Mark each bar's height using the scale (45 = halfway between 40 and 50).
⚠️ Olympiad trap — check the scale

If the grid lines go 0, 5, 10, 15 … each gap is 5, not 1. If they go 0, 20, 40 … a bar ending halfway between 40 and 60 shows 50. Read the scale before reading any bar.

In the ice-cream bar graph, how many more cups of Chocolate than Pista were sold?

  1. 35
  2. 30
  3. 45
  4. 115
Chocolate 75, Pista 40. 75 − 40 = 35. (115 is what you get if you add instead of subtract.)

4Number patterns and sequences

A sequence is a list of numbers that follows a rule. To find the rule, look at the difference between neighbouring numbers first. If that doesn't work, try multiplying or look for special numbers.

SequenceRuleNext
7, 12, 17, 22, …add 5 each time27
90, 81, 72, 63, …subtract 954
3, 6, 12, 24, …multiply by 248
1, 4, 9, 16, 25, …square numbers (1×1, 2×2, 3×3 …)36
1, 3, 6, 10, 15, …add 2, 3, 4, 5 … (triangular numbers)21
3, 4, 7, 12, 19, …add 1, 3, 5, 7, 9 (odd numbers)28
1, 1, 2, 3, 5, 8, …add the two numbers before13
2, 10, 4, 20, 6, 30, …two patterns mixed: 2, 4, 6 … and 10, 20, 30 …8
✏️ Worked example — difference of differences

Find the next number: 4, 6, 10, 16, 24, ___

  1. Differences: 6 − 4 = 2, 10 − 6 = 4, 16 − 10 = 6, 24 − 16 = 8.
  2. The differences go up by 2 each time, so the next difference is 10.
  3. Next number = 24 + 10 = 34.
✏️ Worked example — find the wrong number

Which number is wrong in 5, 10, 20, 40, 80, 150?

  1. Rule: each number is double the one before.
  2. 80 × 2 = 160, not 150. The wrong number is 150 (it should be 160).
Remember: D-M-S — first check the Difference, then Multiplying, then Special numbers (squares, odds, evens, primes, two mixed patterns).

What comes next? 2, 3, 6, 11, 18, ___

  1. 25
  2. 27
  3. 26
  4. 29
Differences are 1, 3, 5, 7 (odd numbers), so the next difference is 9: 18 + 9 = 27.

5Magic squares

In a magic square, every row, every column and both diagonals add up to the same number, called the magic sum.

816357492Every line adds to 15
The numbers 1 to 9: magic sum 15, centre 5.
10A8B7C6DEFind A, B, C, D, E
A puzzle to solve below.
📌 Magic square rules for 3 × 3
  • Magic sum = 3 × the centre number. (Centre 5 → sum 15.)
  • When 1 to 9 are used, the centre must be 5 and the magic sum is 15. (1 + 2 + … + 9 = 45, and 45 ÷ 3 rows = 15.)
  • Two numbers opposite each other through the centre add up to 2 × centre (8 + 2 = 10, 4 + 6 = 10, 1 + 9 = 10, 3 + 7 = 10).
  • Total of all nine numbers = 3 × magic sum.
✏️ Worked example — fill in the magic square

Use the puzzle square above (top row 10, A, 8; middle B, 7, C; bottom 6, D, E).

  1. Centre is 7, so magic sum = 3 × 7 = 21. Check with the diagonal 8 + 7 + 6 = 21 ✓
  2. Top row: 10 + A + 8 = 21 → A = 3.
  3. Diagonal: 10 + 7 + E = 21 → E = 4.
  4. Right column: 8 + C + 4 = 21 → C = 9. Middle row: B + 7 + 9 = 21 → B = 5.
  5. Bottom row: 6 + D + 4 = 21 → D = 11. Check middle column: 3 + 7 + 11 = 21 ✓
💡 Shortcut — start with a full line

Always begin with a row, column or diagonal that has only one blank, or find the centre first and use "magic sum = 3 × centre". Each new number you find opens up another line.

The centre of a 3 × 3 magic square is 8. What is its magic sum?

  1. 8
  2. 16
  3. 24
  4. 72
Magic sum = 3 × centre = 3 × 8 = 24. (72 is the total of all nine numbers: 3 × 24.)

6Missing numbers and number puzzles

Number walls (pyramids)

In a number wall, each brick is the sum of the two bricks just below it. To go down, subtract.

21 11 P 4 Q 3
Brick above = sum of the two bricks below it.
✏️ Worked example — number wall
  1. Top: 11 + P = 21 → P = 10.
  2. 11 sits on 4 and Q: 4 + Q = 11 → Q = 7.
  3. Check: P sits on Q and 3: 7 + 3 = 10 ✓

Consecutive numbers

✏️ Worked example — three in a row

The sum of three consecutive numbers is 84. Find them.

  1. The middle number is the average: 84 ÷ 3 = 28.
  2. The numbers are 27, 28, 29. Check: 27 + 28 + 29 = 84 ✓
  3. Trick: for three consecutive even (or odd) numbers, the middle is still sum ÷ 3, and the others are 2 less and 2 more.

"Who am I?" puzzles

✏️ Worked example — clue by clue (real olympiad question)

Jaya thinks of an even number that is a multiple of 4 and a factor of 56. It is greater than 10 but smaller than 30. What is it?

  1. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56.
  2. Between 10 and 30: 14 and 28.
  3. Multiple of 4? 14 ✗ (14 ÷ 4 is not exact), 28 ✓.
  4. Answer: 28.
✏️ Worked example — swapping digits

I am a 2-digit number. My digits add up to 10. When my digits are swapped, the new number is 36 more than me. Who am I?

  1. Swapping the digits changes a number by 9 × (difference of the digits). 36 ÷ 9 = 4, so the digits differ by 4.
  2. Two digits that add to 10 and differ by 4: 3 and 7.
  3. The swapped number is bigger, so the smaller digit is in the tens place: 37. Check: 73 − 37 = 36 ✓
✏️ Worked example — missing digits in a subtraction

4 □ 7 − 1 8 △ = 2 6 9. Find □ and △.

  1. Turn it into an addition: 1 8 △ + 2 6 9 = 4 □ 7.
  2. Ones: △ + 9 ends in 7 → △ = 8 (8 + 9 = 17), carry 1.
  3. Tens: 8 + 6 + 1 = 15 → □ = 5, carry 1. Hundreds: 1 + 2 + 1 = 4 ✓
  4. So □ = 5, △ = 8: 457 − 188 = 269 ✓

The sum of three consecutive numbers is 84. What is the smallest of them?

  1. 28
  2. 29
  3. 26
  4. 27
Middle = 84 ÷ 3 = 28, so the numbers are 27, 28, 29. The smallest is 27.

7Secret-digit puzzles (cryptarithms) and number logic

In a cryptarithm, digits are hidden behind letters or symbols. The same symbol always means the same digit. Solve them column by column, starting from the ones, and watch the carries.

✏️ Worked example — find ★ + ▲

3 ★ 5 + 2 ▲ 8 = 6 1 3

  1. Ones: 5 + 8 = 13 → write 3, carry 1.
  2. Hundreds: 3 + 2 = 5, but the answer has 6 → there must be a carry of 1 from the tens.
  3. Tens: ★ + ▲ + 1 must end in 1 and make a carry, so ★ + ▲ + 1 = 11.
  4. ★ + ▲ = 10.
✏️ Worked example — repeated digits

△△ + △△ + △△ = 132. What is △?

  1. Three equal numbers make 132, so △△ = 132 ÷ 3 = 44.
  2. △ = 4.
✏️ Worked example — AB + BA

AB and BA are 2-digit numbers and AB + BA = 121. Find A + B.

  1. AB = 10 × A + B and BA = 10 × B + A.
  2. Adding: 11 × A + 11 × B = 11 × (A + B) = 121.
  3. A + B = 121 ÷ 11 = 11. (For example 47 + 74 = 121 ✓, 29 + 92 = 121 ✓.)
💡 Shortcuts for digit puzzles
  • Adding two numbers, the carry into the next column can only be 0 or 1.
  • If a 3-digit + 3-digit sum has 4 digits, the first digit of the answer must be 1.
  • Check your answer by putting the digits back — it takes seconds and catches mistakes.
  • Test the options! In a multiple-choice question, trying each answer is often quickest.
⚠️ Olympiad trap — forgetting the carry

Most wrong answers in cryptarithms come from forgetting a carry. Write the small "1" above the next column every time a column adds up to 10 or more.

AB + BA = 132, where AB and BA are 2-digit numbers. What is A + B?

  1. 12
  2. 11
  3. 13
  4. 6
AB + BA = 11 × (A + B). So 11 × (A + B) = 132, and A + B = 132 ÷ 11 = 12 (e.g. 39 + 93 = 132).

Explore

8Quick recap

Flip cards

Question3 tally gates and 4 lines = ?
Answer15 + 4 = 19
QuestionKey: 1 symbol = 20. What do 4½ symbols show?
Answer80 + 10 = 90
QuestionNext: 1, 4, 9, 16, 25, ?
Answer36 (6 × 6)
QuestionNumbers 1–9 in a magic square: what's in the centre?
Answer5 (magic sum 15)
QuestionMagic sum 30. Centre = ?
Answer30 ÷ 3 = 10
QuestionThree consecutive numbers add to 60. Largest?
AnswerMiddle 20 → 19, 20, 21. Largest 21.
Question□□ × 2 = 66. □ = ?
Answer□□ = 33, so □ = 3
QuestionBricks below are 6 and 9. Brick above?
Answer6 + 9 = 15