Class 8 Ganita Prakash Part 2 Chapter 6 Algebra Play: Complete NCERT Solutions, Think of a Number Tricks, Pyramids, Calendar Magic & Divisibility

Introduction

“Algebra Play” is Chapter 6 of Ganita Prakash Part-II (Class 8). It turns algebra into fun magic! You will discover why “Think of a Number” tricks always give the same answer, how number pyramids work with letter-numbers, calendar magic using 2×2 grids, the largest product from three digits, secret divisibility rules (by 9, 11, 37), and the famous Karim-genie coin puzzle. Every trick is explained with algebra so you can invent your own and amaze friends.

Basic Knowledge Required

• Solving simple linear equations
• Using letter-numbers (variables) to represent unknowns
• Expanding and simplifying expressions
• Divisibility rules (by 9, 11)
• Sum of first n odd numbers and Fibonacci-like sequences

Key Definitions

• Think of a Number Trick: Steps that always end with the same result, no matter the starting number.
• Number Pyramid: Each number is the sum of the two below it; top value is a linear combination of bottom row.
• Calendar Magic: 2×2 grid sum = 4a + 16 → recover original numbers.
• Divisibility Trick: Difference of a number and its reverse is always divisible by 9.

Important Formulas Used

1. Think-of-a-number: Final = constant (e.g., x → 2x + 4 → x + 2 → 2)
2. Date trick: Final = 100M + D + 165 → M & D from last two digits after –165
3. Pyramid (3 rows): Top = a + 2b + c
4. Largest product (p < q < r): qp × r (largest digit as multiplier)
5. Reverse digits difference: 9(b – a)
6. Karim-genie: After 3 rounds with cost 8: 8x – 24 = 8 → x = 4 coins start

6.2 Thinking about ‘Think of a Number’ Tricks

Original trick always ends with 2 Algebra: x → 2x → 2x+4 → x+2 → (x+2) – x = 2

Figure it Out (Page 136)

1. To get 3: Change “Add four” to “Add six” (or adjust any step by +2). To get 5: Add eight instead of four.
2. More complicated: Multiply by 3, add 6, divide by 3, subtract original → always 2.

Date trick Steps: 5M → 5M+6 → 20M+24 → 20M+33 → 100M+165 → 100M+165+D Subtract 165 → last two digits = D, before = M.

Figure it Out (Page 137) Mukta’s 1390 → 1390 – 165 = 1225 → 25 December. (i) 1269 – 165 = 1104 → 4 November (ii) 394 – 165 = 229 → 29 February (or adjust month) (iii) 296 – 165 = 131 → 31 January

6.3 Number Pyramids

Fill the pyramids Figure it Out (Page 138–139)

1. Top-left pyramid: 6+2=8 Middle: 3+4+3=10, 4+3=7 → top 17 Right: 5+4+5+0=14, etc. → complete step-by-step sums.
2. Bottom-up or top-down: 10 – 4 = 6; 4 – 1 = 3; 6 – 3 = 3.
3. 60 pyramid: a+b=60, 12+c=a, c+8=b → 20+2c=60 → c=20, a=32, b=28.

Relationship Top = sum of bottom row with coefficients (1,2,1 for 3-row).

Figure it Out (Page 140)

1. Bottom 4 13 8 → top 4+2×13+8=38 7 11 3 → 7+2×11+3=32 10 14 25 → 10+2×14+25=63
2. 4-row top = a + 3b + 3c + d
3. Bottom 8 19 21 13 → top 8+3×19+3×21+13=122 7 18 19 6 → 7+3×18+3×19+6=106 9 7 5 11 → 9+3×7+3×5+11=50

4–6. Fibonacci in pyramid: every entry is a Fibonacci number; top is also Fibonacci (sum with binomial coefficients). For n rows, top is the (2n–1)th Fibonacci number.

6.4 Fun with Grids

Calendar Magic 2×2 sum = 4a + 16. Example sum 40 → a=6 (grid 6 7 / 13 14). Sum 36 → a=5 (5 6 / 12 13).

Figure it Out Create own: 3×3 sum = 9a + 36 → recover numbers.

Algebra Grids Figure it Out (Page 142) First grid: □=9, ●=5 (solve row equations). Second: ◆=4, ●=3 (similar substitution).

6.5 The Largest Product

Digits 2,3,5 Six options → group by multiplier → 32×5=160 is largest.

General rule (p<q<r): Largest = qp × r

Figure it Out (Page 143)

1. 1,3,7 → 73×1=73; 71×3=213; 31×7=217 → 31×7=217 largest.
2. 3,5,9 → 95×3=285; 93×5=465; 53×9=477 → 53×9=477 largest.

6.6 Decoding Divisibility Tricks

Reverse digits difference Always ÷9: (10b+a) – (10a+b) = 9(b–a).

Figure it Out (Page 145–146)

1. Quotient = |b–a| (difference of digits).
2. Sum always ÷11: (10a+b) + (10b+a) = 11(a+b).
3. 3-digit cycle sum = 111(a+b+c) ÷37 (and ÷3).
4. abcabc = 1001×abc → 1001=7×11×13 → ÷7×11×13 = abc.
5. Shrines: Start with 8 flowers → each shrine 8 (doubling works backward).
6. Horses 10, hens 45 (55 heads, 150 legs).
7. Daughter 3 years (mother 15 now).
8. Gauri 3, Naina 6 cows.
9. (i) ₹70 per dosa; (ii) 150 dosas.
10. Each fraction = 1/2 (sum odd numbers formula).

Karim & Genie Start with 4 coins → after 3 rounds exactly 8 left.

15 Most Asked FAQs – Algebra Play (Click to Expand)

1. Original “Think of a Number” trick always ends with?

Answer: 2 (algebra: x → 2x+4 → x+2 → 2).

2. Date trick final answer 291 → date?

Answer: 26 January (291–165=126 → M=1, D=26).

3. 60 pyramid bottom 12 ? 8 → top numbers?

Answer: 32 28 (c=20, a=32, b=28).

4. Calendar 2×2 sum 40 → grid numbers?

Answer: 6 7 / 13 14 (4a+16=40 → a=6).

5. Largest product 2,3,5 in □□×□?

Answer: 32×5=160 (qp×r rule).

6. Reverse digits difference always divisible by?

Answer: 9 (9(b–a)).

7. abcabc ÷7÷11÷13 = ?

Answer: Original abc (1001=7×11×13).

8. Horses + hens = 55 heads, 150 legs → horses?

Answer: 10 horses, 45 hens.

9. Mother 5× daughter age; in 6 years 3× → daughter age?

Answer: 3 years (mother 15).

10. Gauri & Naina cows: twice +3 equal → numbers?

Answer: Gauri 3, Naina 6.

11. Dosa cart ₹5000 rent + ₹10/dosa; 100 dosas profit ₹2000 → price?

Answer: ₹70.

12. Karim-genie start coins?

Answer: 4 coins (8x–24=8).

13. Pyramid top for bottom 4 13 8?

Answer: 38 (a+2b+c).

14. Sum of reverse digits always divisible by?

Answer: 11 (11(a+b)).

15. Fibonacci in n-row pyramid top?

Answer: (2n–1)th Fibonacci number.