Class 6 Number Play Ganita Prakash

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Chapter 3: Class 6 Number Play Solutions

Chapter 3: Class 6 Number Play Solutions

This chapter is all about having fun with Chapter 3: Class 6 Number Play Solutions! We’ll discover hidden patterns, solve puzzles, and learn strategies for games, all using the numbers we see every day.

3.1 Numbers can Tell us Things

The Height Puzzle (Page 56)

The Rule: Each child says a number that is equal to the number of their immediate neighbours (the person on their left and right) who are taller than them.

Let’s use numbers 1 (shortest) to 5 (tallest) to represent the children’s heights.

  1. Can the children at the ends say ‘2’?
    No. A child at an end only has one neighbour. The maximum number they can say is ‘1’ (if that one neighbour is taller).
  2. Can we arrange them so all say ‘0’?
    Yes. If they are arranged in decreasing order of height (e.g., 5, 4, 3, 2, 1). For any child, their neighbours (if any) will always be shorter, so everyone will say ‘0’.
  3. Can two children standing next to each other say the same number?
    Yes. Consider the arrangement: 3, 5, 2, 4, 1.
    • The child with height 3 (neighbor 5) says ‘1’.
    • The child with height 5 (neighbors 3, 2) says ‘0’.
    • The child with height 2 (neighbors 5, 4) says ‘2’.
    • The child with height 4 (neighbors 2, 1) says ‘0’.
    • The child with height 1 (neighbor 4) says ‘1’.
    In this case, the two children at the ends both say ‘1’. So it is possible.
  4. With 5 children, can four say ‘1’ and one say ‘0’?
    Yes. The child who says ‘0’ must be the tallest, since they have no taller neighbours. To make the other four children say ‘1’, the tallest child must be placed strategically. Arrangement: 3, 4, 5, 2, 1.
    • Child 3 (neighbor 4) says ‘1’.
    • Child 4 (neighbor 5) says ‘1’.
    • Child 5 (neighbors 4, 2) says ‘0’.
    • Child 2 (neighbor 5) says ‘1’.
    • Child 1 (neighbor 2) says ‘1’.
    This arrangement gives the sequence 1, 1, 0, 1, 1.
  5. Is the sequence 1, 1, 1, 1, 1 possible?
    No. The tallest child in the group will never have a taller neighbour, so they must always say ‘0’. A sequence of all ‘1’s is impossible.
  6. Is the sequence 0, 1, 2, 1, 0 possible?
    Yes. This arrangement works: 5, 3, 1, 2, 4.
    • Child 5 (neighbor 3) says ‘0’.
    • Child 3 (neighbors 5, 1) says ‘1’.
    • Child 1 (neighbors 3, 2) says ‘2’.
    • Child 2 (neighbors 1, 4) says ‘1’.
    • Child 4 (neighbor 2) says ‘0’.
  7. How can we arrange five children so the maximum number of children say ‘2’?
    The maximum number of children who can say ‘2’ is two. A child must be a “valley” (shorter than both neighbours) to say ‘2’. With 5 children, we can create at most two valleys. Arrangement: 3, 1, 4, 2, 5.
    • Child 1 (neighbors 3, 4) says ‘2’.
    • Child 2 (neighbors 4, 5) says ‘2’.

3.2 Supercells

Puzzles on Supercells (Pages 57-58)

The Rule: A “supercell” is a cell whose number is larger than all of its adjacent neighbors (top, bottom, left, right).

  • Q6. Can you fill a table with no supercells?
    Yes. Arrange the numbers in increasing order, for example, in a snake-like pattern. This ensures every number (except the largest) has a larger neighbour.
  • Q7. Is the largest number always a supercell? Is the smallest number ever a supercell?
    The largest number in the entire table will always be a supercell because it has no larger neighbours. The smallest number can never be a supercell because all its neighbours will be larger.
  • Q8. Fill a table so the second largest number is not a supercell.
    Yes. Simply place the largest number adjacent to the second-largest number. The second-largest number will then have a larger neighbour, so it won’t be a supercell.

3.5 Pretty Palindromic Patterns

A palindrome is a number that reads the same forwards and backwards, like 121, 4884, or 66. The “reverse-and-add” method is a fun way to create them:

  1. Take any number (e.g., 48).
  2. Reverse its digits (84).
  3. Add them together: 48 + 84 = 132.
  4. If the result is not a palindrome, repeat: 132 + 231 = 363. Now it’s a palindrome!

3.6 The Magic Number of Kaprekar

Puzzle Time (Page 62)

Clues: I am a 5-digit odd palindrome. My tens digit (‘t’) is double my units digit (‘u’). My hundreds digit (‘h’) is double my tens digit (‘t’). Who am I?

Let’s decode the clues:

  • The number is a 5-digit palindrome: h t u t h wait, the format is u t h t u. Let’s check the book. The place values are tth, th, h, t, u. So the palindrome is `u t h t u`. No, that’s not right. A 5-digit palindrome is of the form `a b c b a`. Let’s use the letters from the puzzle: u, t, h. So the number is `x y h y u`. The units digit is u, tens is y, hundreds is h. The puzzle says “My ‘t’ digit is double my ‘u’ digit.” This is tens and units. “My ‘h’ digit is double my ‘t’ digit”. Let’s assume the digits from right to left are u, t, h, t, u.
  • The number is odd, so the units digit u must be 1, 3, 5, 7, or 9.
  • t = 2 × u
  • h = 2 × t = 2 × (2 × u) = 4 × u

Now, let’s test values for u:

  • If u = 1, then t = 2×1 = 2, and h = 4×1 = 4. All are single digits. The number is 12421.
  • If u = 3, then t = 2×3 = 6, but h = 4×3 = 12. A digit cannot be 12. So this doesn’t work.

The only possible answer is 12421.

Kaprekar’s Constant (Page 63)

For any 4-digit number with at least two different digits, if you repeatedly follow these steps, you will always reach the magic number 6174, the Kaprekar constant.

What about 3-digit numbers? If you try the same process, you will always reach the number 495.

3.12 Games and Winning Strategies

Game #1: The 21 Game (Page 71)

Rules: Players take turns adding 1, 2, or 3 to the previous number. The first player to say 21 wins.

This is a strategy game where you can guarantee a win by controlling certain “magic numbers”. The key is to work backwards from the target number, 21.

  • To say 21, you need your opponent to give you a number between 18 and 20. To force this, you must say 17.
  • To say 17, you must first say 13 (since 13 + (1,2, or 3) gives 14, 15, or 16, allowing you to say 17).
  • To say 13, you must first say 9.
  • To say 9, you must first say 5.
  • To say 5, you must first say 1.

Winning Strategy: The first player can always win! They must say the magic numbers: 1, 5, 9, 13, 17, and finally 21. The pattern is based on the number 4 (the sum of the smallest and largest possible additions, 1+3).

Game #2: The 99 Game (Page 71)

Rules: Players take turns adding a number from 1 to 10. The first player to reach 99 wins.

This game has a similar strategy, but the key number is 11 (since you can add from 1 to 10).

  • To win, you must say 99. To do this, you must say 88 first. (If you say 88, your opponent can add any number from 1 to 10, giving a sum from 89 to 98. In every case, you can add the correct number to reach 99).
  • The other magic numbers to say are multiples of 11, working backwards: 77, 66, 55, 44, 33, 22, 11.

Winning Strategy: The second player can always win!
If Player 1 says a number X (from 1 to 10), Player 2 should respond by saying a number that makes the total sum 11. (Player 2 adds 11 - X). Then, on every subsequent turn, Player 2 should make the total sum the next multiple of 11. By doing this, Player 2 will say 88, and will be able to say 99 on the next turn to win.

SUMMARY

  • Numbers are used for more than just counting; they help us find patterns, solve puzzles, and win games.
  • Thinking logically about a problem and breaking it down into steps is called computational thinking.
  • A palindrome is a number that reads the same forwards and backwards (e.g., 585).
  • Kaprekar’s Constant (6174) is a fascinating result of a simple process of rearranging and subtracting digits.
  • Many games with numbers have a hidden winning strategy that can be discovered by working backwards from the goal.
  • Some problems in mathematics, like the Collatz Conjecture, are very easy to state but remain unsolved to this day.