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CBSE · Class 7 · Mathematics

Chapter 6: Number Play

5 concepts · 40 practice questions with answers · Free to read

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Patterns in sums of numbers

From daily life

Stacking 1, 3, 5 and 7 bricks in L-shaped layers builds a perfect 4 × 4 square of 16 bricks.

Simple definition

Adding numbers in a pattern often gives a quick shortcut. Three consecutive numbers add to 3 × the middle one; the first n odd numbers add to n × n.

1+ 3+ 5+ 7= 16 = 4 × 4

In the figure: Odd-number layers 1, 3, 5, 7 build the square 4 × 4 = 16.

Rule 1: Three consecutive numbers — Sum = 3 × the middle number.

  • Easy: 7 + 8 + 9 = 24 = 3 × 8.
  • Medium: 29 + 30 + 31 = 90.
  • Exam-type: Sum 51 → middle 51 ÷ 3 = 17 → numbers 16, 17, 18.

Rule 2: Sum of first n odd numbers — Count the odd numbers and multiply that count by itself.

  • Easy: 1 + 3 + 5 = 9 = 3 × 3.
  • Medium: First 9 odd numbers add to 81.
  • Exam-type: Sum 144 → 12 × 12 → the first 12 odd numbers.

Rule 3: Adding 1, 2, 3, … — Pair the first and last, the second and second-last: every pair gives the same sum.

  • Easy: 1 + 2 + 3 + 4 = 10 (two pairs of 5).
  • Medium: 1 + 2 + … + 10 = 5 pairs of 11 = 55.
  • Exam-type: 1 + 2 + … + 20 = 10 pairs of 21 = 210.
Remember

Consecutive means one after another with a gap of 1.

Common mistake

Using 3 × middle for numbers that are not consecutive, such as 5, 7, 9 (gap 2; here it still works, but 5, 8, 9 does not).

Shortcut

Check the gaps before using the shortcut. Memory clue: "Middle times 3"; "odd layers make squares."

Spot the difference

PatternRuleExample
3 consecutive3 × middle4 + 5 + 6 = 15
first n oddn × n1 + 3 + 5 + 7 = 16

Practice questions

Easy
  1. 1Find 14 + 15 + 16.
  2. 2Find the sum of the first 6 odd numbers.
  3. 3Find 1 + 2 + 3 + 4 + 5.
Medium
  1. 4Three consecutive numbers add to 63. Find them.
  2. 5The first how many odd numbers add to 100?
  3. 6Find 1 + 2 + … + 30.
Challenging
  1. 7Can 3 consecutive numbers add to 100? Why?
  2. 8Add 51 + 52 + … + 60 using pairs.
Show answers (try them yourself first)
  1. 145
  2. 236
  3. 315
  4. 420, 21, 22
  5. 510
  6. 6465
  7. 7No: 100 is not a multiple of 3
  8. 8555 (5 pairs of 111)

Parity: even and odd rules

From daily life

Pairs of shoes: 8 shoes make 4 pairs with none left, but 9 shoes leave one odd shoe.

Simple definition

A number is even if it can be paired with none left over, and odd if one is left. Being even or odd is called parity. Parity helps decide whether a sum is possible without adding.

even: all pairedodd: one left over

In the figure: Six dots pair completely; seven dots leave one red dot alone.

Rule 1: Sum of two numbers — even + even = even; odd + odd = even; even + odd = odd.

  • Easy: 12 + 8 = 20 (even).
  • Medium: 7 + 9 = 16 (even).
  • Exam-type: Two consecutive numbers always add to an odd number.

Rule 2: Many odd numbers — An even count of odd numbers adds to even; an odd count adds to odd.

  • Easy: 3 odd numbers: 1 + 3 + 5 = 9 (odd).
  • Medium: 4 odd numbers: 1 + 3 + 5 + 7 = 16 (even).
  • Exam-type: Can 5 odd cards add to 30? No: 5 odd numbers give an odd sum.

Rule 3: Products — A product is even if any factor is even; odd only if all factors are odd.

  • Easy: 6 × 7 = 42 (even).
  • Medium: 5 × 9 = 45 (odd).
  • Exam-type: 3 × 5 × 7 × 8 is even because of the 8.
Remember

Parity is decided by the last digit: 0, 2, 4, 6, 8 mean even.

Common mistake

Thinking that odd + odd = odd. It is even (3 + 5 = 8).

Shortcut

Pair up the objects in your mind; a leftover means odd. Memory clue: "Odd + odd = even (two leftovers make a pair)."

Spot the difference

SumParityExample
even + eveneven4 + 6 = 10
odd + oddeven3 + 5 = 8
even + oddodd4 + 5 = 9

Practice questions

Easy
  1. 1Is 7 + 12 even or odd?
  2. 2Is the sum of 6 odd numbers even or odd?
  3. 3Is 9 × 11 even or odd?
Medium
  1. 4Can 4 odd numbers add to 33?
  2. 5Two friends are one year apart in age. Can their ages add to 50?
  3. 6Is 3 × 5 × 7 × 8 even or odd?
Challenging
  1. 77 cards each carry an odd number. Can they add to 40?
  2. 8Is the sum of 5 even numbers and 3 odd numbers even or odd?
Show answers (try them yourself first)
  1. 1Odd
  2. 2Even
  3. 3Odd
  4. 4No: 4 odd numbers give an even sum
  5. 5No: consecutive ages add to an odd number
  6. 6Even
  7. 7No: 7 odd numbers give an odd sum
  8. 8Odd

Patterns in number grids

From daily life

On a calendar, moving one box right adds 1 and moving one row down adds 7.

Simple definition

In a grid where each row has 7 numbers, the numbers follow fixed steps: +1 across, +7 down. These steps explain many patterns such as equal diagonal sums and the plus-shape sum = 5 × centre.

89101516172223249+15+16+17+23= 80= 5 × 16

In the figure: The plus shape 9, 15, 16, 17, 23 adds to 80 = 5 × 16.

Rule 1: Steps in the grid — Right = +1, left = –1, down = +7, up = –7.

  • Easy: From 10 one step down is 17.
  • Medium: From 23 two steps up is 9.
  • Exam-type: From 5 go 2 down and 1 right: 5 + 14 + 1 = 20.

Rule 2: Equal diagonal sums — In any 2 × 2 box the two diagonals have the same sum.

  • Easy: Box 8, 9, 15, 16: 8 + 16 = 9 + 15 = 24.
  • Medium: Box 3, 4, 10, 11: 3 + 11 = 4 + 10 = 14.
  • Exam-type: Box 12, 13, 19 and a missing corner: 13 + 19 = 32, so the corner is 32 – 12 = 20.

Rule 3: Plus-shape sum — Plus sum = 5 × the centre number.

  • Easy: Centre 16: sum = 80.
  • Medium: Centre 11: sum = 55.
  • Exam-type: Plus sum 100 → centre 100 ÷ 5 = 20.
Remember

The four neighbours of a centre c are c – 7, c – 1, c + 1, c + 7; their changes cancel in pairs.

Common mistake

Using 5 × centre for a box that touches the edge of the calendar (no full plus fits).

Shortcut

Check that all five boxes exist before using the shortcut. Memory clue: "Plus = five times the heart."

Spot the difference

MoveChangeFrom 16
right+117
down+723
up-left–88

Practice questions

Easy
  1. 1In the calendar grid, which number is one row below 12?
  2. 2Find the plus-shape sum with centre 18.
  3. 3Do the diagonals of box 5, 6, 12, 13 have equal sums?
Medium
  1. 4The plus sum is 135. Find the centre.
  2. 5From 4 go 3 down and 2 right. Where do you land?
  3. 6In a box 9, 10, 16, ?, find the missing corner.
Challenging
  1. 7Can the plus sum be 72? Why?
  2. 8Three numbers in a row add to 51. Find them.
Show answers (try them yourself first)
  1. 119
  2. 290
  3. 3Yes: 5 + 13 = 6 + 12 = 18
  4. 427
  5. 54 + 21 + 2 = 27
  6. 617 (10 + 16 – 9)
  7. 7No: 72 is not a multiple of 5
  8. 816, 17, 18
Live figure

Patterns in number grids

In the app you can move this figure with your finger.

Open in the app

The Virahanka (Fibonacci) sequence

From daily life

You climb stairs taking 1 or 2 steps at a time. For 4 stairs there are 5 different ways; for 5 stairs, 8 ways.

Simple definition

The sequence 1, 1, 2, 3, 5, 8, 13, 21, … in which each term is the sum of the two terms before it is the Virahanka (Fibonacci) sequence. Virahanka found it while counting rhythm patterns made of short (1) and long (2) beats.

11235813+++++each term = sum of the two before it

In the figure: Terms 1, 1, 2, 3, 5, 8, 13: 8 + 13 = 21 comes next.

Rule 1: Build the sequence — Add the last two terms to get the next.

  • Easy: 1, 1, 2, 3, 5 → next 8.
  • Medium: Next after 13, 21 is 34.
  • Exam-type: Starting with 2, 3: 2, 3, 5, 8, 13, 21 (same rule, new start).

Rule 2: Find a missing term — A term is the sum of its two neighbours before it, so subtract to go back.

  • Easy: …, 5, ?, 13: ? = 13 – 5 = 8.
  • Medium: …, ?, 8, 13: ? = 13 – 8 = 5.
  • Exam-type: …, ?, 21, 34 → ? = 13; and before it 21 – 13 = 8.

Rule 3: Hidden patterns — Every third term is even; the sum of the first few terms is one less than a later term.

  • Easy: Even terms: 2, 8, 34, … (3rd, 6th, 9th).
  • Medium: 1 + 1 + 2 + 3 + 5 = 12 = 13 – 1.
  • Exam-type: 1 + 1 + 2 + 3 + 5 + 8 + 13 = 33 = 34 – 1.
Remember

Count rhythm or stair ways by adding the two previous counts.

Common mistake

Adding the last term and the term before the previous one (skipping one). Use the two immediate terms.

Shortcut

Write the sequence in a row and put a + sign between the last two. Memory clue: "Two behind, add, and go ahead."

Spot the difference

StairsWaysHow
11—
221+1, 2
332 + 1
453 + 2
585 + 3

Practice questions

Easy
  1. 1Write the next term: 1, 1, 2, 3, 5, 8, __.
  2. 2Find the missing term: 2, 3, 5, __, 13.
  3. 3How many ways to climb 3 stairs with steps of 1 or 2?
Medium
  1. 4Continue 3, 4, 7, 11 for two terms.
  2. 5How many ways to climb 6 stairs?
  3. 6Which of 8, 21, 34, 55 are even?
Challenging
  1. 7Find the sum 1 + 1 + 2 + 3 + 5 + 8 using the hidden pattern.
  2. 8Terms ?, ?, 21, 34, 55: find the first two.
Show answers (try them yourself first)
  1. 113
  2. 28
  3. 33
  4. 418, 29
  5. 513
  6. 68 and 34
  7. 720 (= 21 – 1)
  8. 88 and 13

Digits in disguise (letter puzzles)

From daily life

A secret code replaces digits by letters. Cracking it needs logic about the last digit and the carry.

Simple definition

In a letter puzzle each letter stands for one digit, different letters stand for different digits, and the first digit of a number is not 0. Solve column by column from the ones place, remembering the carry.

A B+ A B1 5 6B + B = 6→ B = 8A = 778 + 78 = 156

In the figure: AB + AB = 156: units B + B ends in 6, so B = 8; then 2A + 1 = 15 gives A = 7.

Rule 1: Ground rules — One letter = one digit; the leading letter is not 0; use carries between columns.

  • Easy: A + A = 12 → A = 6.
  • Medium: A + 5 = 11 → A = 6.
  • Exam-type: 2 × K ends in 4 → K = 2 or 7.

Rule 2: Solve AB + AB — Use the ones column first, then the tens column with the carry.

  • Easy: AB + AB = 156 → 78 + 78.
  • Medium: AB + AB = 92 → 46 + 46.
  • Exam-type: AB + AB = 128 → 64 + 64.

Rule 3: A number and its reverse — AB + BA is always 11 × (A + B).

  • Easy: 25 + 52 = 77 = 11 × 7.
  • Medium: 38 + 83 = 121 = 11 × 11.
  • Exam-type: AB + BA = 143 → A + B = 13 (for example 49 + 94).
Remember

Always check the final answer by adding back.

Common mistake

Giving two letters the same digit. Different letters need different digits.

Shortcut

Start with the column that has the fewest unknowns. Memory clue: "Ones first, carry next."

Spot the difference

StepQuestionUse
1Ones digit?B + B ends in 6
2Carry?8 + 8 = 16 → carry 1
3Tens digit?2A + 1 = 15

Practice questions

Easy
  1. 1A + A = 14. Find A.
  2. 2If K + 8 = 15, find K.
  3. 3Find 25 + 52.
Medium
  1. 4AB + AB = 92. Find AB.
  2. 5AB + AB = 128. Find AB.
  3. 6AB + BA = 99 and A = 3. Find B.
Challenging
  1. 7AB + BA = 143. What is A + B?
  2. 8AB + AB = 144. Find AB.
Show answers (try them yourself first)
  1. 17
  2. 27
  3. 377
  4. 446
  5. 564
  6. 66 (A + B = 9)
  7. 713
  8. 872