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CBSE Class 6 · Mathematics

Number Play

Official NCERT chapter from Ganita Prakash (book code fegp1). ExamMaster notes are original teaching at CBSE Class 6 depth.

This lesson follows the official chapter “Number Play” in Ganita Prakash. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 6
  • Easy level
  • 12 concepts

1Numbers can Tell us Things

A number can be even or odd, a multiple, a code for a house. It tells a fact only after you name the test you ran.

A number sitting alone is mute until the test is named.

Figure. Numbers can tell a fact you could not see from a name. 135 cm against 128 cm says Zoya is taller — the number is the message, not decoration. A class roll, a house number, and a bus number work the same way.

How it works

  1. Name the testEven, multiple of 5, house number…
  2. Run it on the numberPass or fail.
  3. Say what it told youThe fact, not a vibe.

2Supercells

A supercell here is a highlighted cell in a grid of numbers that wins a local compare — larger than its neighbours, or matching a stated rule.

Colouring a cell because it looks pretty is not a supercell.

Figure. A supercell holds the sum of the cells it covers. 13 is 5 + 8; 7 is 3 + 4. You check a supercell by adding, not by guessing a prettier number.

How it works

  1. Read the compare-ruleLargest among neighbours, or another stated test.
  2. Check each neighbourEvidence.
  3. Mark only the cells that passThose are the supercells.

3Patterns of Numbers on the Number Line

Even numbers sit every other mark: 0, 2, 4, … Odds sit on the leftover marks. A hop of 5 lands on multiples of 5.

A number line with no hop named is just a ruler.

Figure. A pattern on the number line is a jump you can name. Here every mark is 4 more than the last. From 12 the next mark is 16, not a guess — the same +4 that made 8 from 4.

How it works

  1. Name the hop2, or 5.
  2. Start at a mark0, or 1.
  3. Land only on the hopThat is the pattern.

4Playing with Digits

Digits can be rearranged, summed, or swapped by place. 257 and 572 use the same digits; the houses make different values.

Rearranging digits is a new number, not a costume on the same value.

Figure. The digit is allowed only after you name its house. 3 in tens is 30; the same stroke in ones is 3. That is why 32 and 23 are different numbers — swapping houses is not a font change.

How it works

  1. List the digits2, 5, 7.
  2. Place them in housesA new numeral.
  3. Read the new valueDifferent houses, different number.

5Pretty Palindromic Patterns

A palindrome reads the same forward and back: 121, 3443. The test is the digit sequence, not a pretty font.

1221 is a palindrome; 1223 is not “almost” one.

Figure. A palindrome reads the same forwards and backwards. 1221 folds on the dashed midline: the outer 1s match and the inner 2s match. 1231 fails the fold — that is the test, not a pretty look.

How it works

  1. Write the digits left to rightThe sequence.
  2. Write them right to leftThe reverse.
  3. Demand they matchThen it is a palindrome.

Palindrome test

Is 2552 a palindrome? Is 2553?

  • 2552 reverse2552
  • 2552yes
  • 2553 reverse3552
  • 2553no

Pro tip. Exact reverse match. Almost is a miss.

6The Magic Number of Kaprekar

Kaprekar’s routine on four digits (not all equal): arrange largest, arrange smallest, subtract, repeat. Many seeds reach 6174. The routine is the maths; “magic” is the nickname.

Do not skip the subtract. A pretty 6174 without the steps is a label.

Figure. Kaprekar's routine: rearrange a four-digit number largest-to-smallest and smallest-to-largest, then subtract. 7641 − 1467 = 6174, and 6174 maps to itself. Most four-digit starts reach this same 6174.

How it works

  1. Form largest and smallest from the digitsTwo numbers.
  2. SubtractA new four-digit (pad zeros if needed).
  3. Repeat until it staysOften 6174 for four distinct-enough digits.

7Clock and Calendar Numbers

A clock wraps at 12 (or 24). A calendar wraps months and weeks. Adding 8 hours to 9 o’clock is not 17 on a 12-hour face without a wrap.

Treat wrap as a remainder after the cycle length.

Figure. Clock numbers are a cycle of 12, not a line that keeps growing. At 3:00 the minute hand is on 12 and the hour hand is on 3 — a right angle. The same instant is 15:00 on a 24-hour calendar clock.

How it works

  1. Name the cycle12 hours, 7 weekdays.
  2. Add or count along the cycleThen wrap.
  3. Write the landing nameA clock or weekday.

8Mental Math

Mental math is a split you can hold: 47+38 as 47+30+8, or 25×16 as 25×10 + 25×6. The split must rebuild the original.

A guess that you cannot rebuild is not mental math.

Figure. Mental math here is a split you can say: 25 × 12 = 25 × 10 + 25 × 2 = 250 + 50 = 300. The split uses place value, not a new trick. A split you cannot say out loud is not yet mental math.

How it works

  1. Split one numberA friendly piece.
  2. Compute the piecesKnown facts.
  3. Add the pieces backThe original product or sum.

9Playing with Number Patterns

A play-pattern still needs a rule you can say. Alternating +2 and −1 is a rule. Scribbling digits is not play in this chapter.

Write the rule before you enjoy the pattern.

Figure. Triangular numbers add one more each time: 1, then 1+2=3, then 3+3=6, then 6+4=10. The next jump is +5, so T5 = 15. Naming the jump is the pattern; copying the last height is not.

How it works

  1. Say the ruleA sentence.
  2. Generate two more termsA check.
  3. If you cannot, it was not a ruleA list.

10An Unsolved Mystery — the Collatz Conjecture

Collatz rule: if even, divide by 2; if odd, 3n+1. Then repeat. The conjecture says every positive start eventually reaches 1. It is unproved — a mystery, not a homework “always” you must pretend is settled.

A chain that reached 1 for 17 does not prove every start.

Figure. Collatz rule: even → n/2, odd → 3n+1. From 6 the path is 6, 3, 10, 5, 16, 8, 4, 2, 1. Nobody has proved every start reaches 1 — the chapter shows the rule, not a finished theorem.

How it works

  1. Pick a startA positive integer.
  2. Apply even/odd ruleA next term.
  3. Stop at 1 or after a named number of stepsA case, not a proof of all n.

Collatz from 6

Start at 6. Write the next four terms using the Collatz rule.

  • 6 even6÷2=3
  • 3 odd3×3+1=10
  • 10 even5
  • 5 odd16

Pro tip. Even → ÷2. Odd → 3n+1. One start is a case, not a proof.

11Simple Estimation

Estimate by nearby friendly numbers: 198×4 is near 200×4=800. Then compute or sense-check.

An estimate is a neighbour, not a replacement when the exact is asked.

Figure. Estimate 198 + 403 by walking each addend to a nearby ten: 200 + 400 = 600. The exact sum is 601, one away. The figure shows the rounding, not two bars that would look the same height.

How it works

  1. Round to a nearby friendly number198→200.
  2. Compute that800.
  3. Say “about” and, if needed, compute exactlyTwo numbers, two labels.

12Games and Winning Strategies

A strategy is a plan that uses the arithmetic (leave a multiple of 3, take 1 or 2). The move is still a counted take. A plan that miscounts is a broken strategy.

Hoping is not a strategy.

Figure. A take-1-2-or-3 game on 21: the winning leaves are multiples of 4 (0, 4, 8, 12, 16, 20). From 21 you take 1 and leave 20. After that, answer k with 4 − k. 21 is not a safe leave — that is the start, not a win.

How it works

  1. Name the legal movesThe rules.
  2. Name a target leftover if you have oneA plan.
  3. Compute the take as subtractionArithmetic first.
Collatz from one start reaching 1
  1. Is a case, not a proof for every n
  2. Settles the conjecture
  3. Means the rule failed

The conjecture is still a mystery. One chain is evidence for that start only.

Notes

  • The official chapter title is “Number Play”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • Palindrome: digits = reverse(digits).
  • Clock wrap: landing = remainder after the cycle length.
  • Collatz: even \rightarrow n/2; odd \rightarrow 3n+1 (one start is a case).

Recap

Hold these pegs from the official chapter “Number Play”. The wording is ExamMaster’s teaching, not a textbook recap.

Numbers can Tell us Things
A number can be even or odd, a multiple, a code for a house.
Supercells
A supercell here is a highlighted cell in a grid of numbers that wins a local compare — larger than its neighbours, or matching a stated rule.
Patterns of Numbers on the Number Line
Even numbers sit every other mark: 0, 2, 4, … Odds sit on the leftover marks.
Playing with Digits
Digits can be rearranged, summed, or swapped by place.
Pretty Palindromic Patterns
A palindrome reads the same forward and back: 121, 3443.
The Magic Number of Kaprekar
Kaprekar’s routine on four digits (not all equal): arrange largest, arrange smallest, subtract, repeat.

Practise Number Play

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