NCERT Solutions Ganita Prakash (Part 2) Chapter 6 .2 Thinking about ‘Think of a Number’ Tricks — In-text Questions

Book page 1366 Updated on2026-09-05

Q1.

How would you change this game to make the final answer 3? What about 5?

Answer

Change only step 3 — the number you add. Add 6 to finish at 3, and add 10 to finish at 5.

Think of a number: x
Double it: 2x
Add 6: 2x + 6
Divide by two: x + 3
Subtract the original number: 3

Add 10 instead: 2x + 10 → x + 5 → 5

Why it happens: the number you add is put in while there are two x’s, and step 4 halves everything. So the constant that survives is always half of what you added. To finish at any value k, add 2k:
2x → 2x + 2k → x + k → k.

Check it yourself: start with 11 and add 6 — 22, 28, 14, and 14 – 11 = 3. ✓

Q2.

Can you come up with more complicated steps that always lead to the same final value?

Answer

Yes. The recipe is simple: write the expression next to every step, and make sure the last step removes every x that is left.

Think of a number: x
Multiply by 3: 3x
Add 12: 3x + 12
Divide by 3: x + 4
Add 6: x + 10
Subtract the number you thought of: 10

A longer one, with a detour that changes nothing — it still ends at 10:

Think of a number: x
Multiply by 4: 4x
Add 40: 4x + 40
Halve it: 2x + 20
Subtract 20: 2x
Halve it again: x
Add 10: x + 10
Subtract the number you thought of: 10

Why it happens: a trick works when the expression just before the last step is (one x) + (a constant). Whatever you do in between — multiply, add, subtract, divide — you only need the amount of x to come back to exactly 1 before the final subtraction. Multiplying by 4 and later dividing by 4 does exactly that, and the 40 that rode along becomes 10.

Try This: design one where the final subtraction is not the last step, for example: think of x, treble it, add 15, divide by 3, subtract x, then multiply by 2 — you get 10 every time.

Q3.

How did Shubham figure out the date chosen by Mukta?

Answer

He undid the steps with algebra. Let the month be M and the day be D, and follow the instructions on the letter-numbers:

Multiply the month by 5: 5M
Add 6: 5M + 6
Multiply by 4: 20M + 24
Add 9: 20M + 33
Multiply by 5: 100M + 165
Add the day: 100M + 165 + D

Mukta’s answer was 291, so

100M + 165 + D = 291
100M + D = 291 – 165 = 126 (subtracting 165 from both sides)
So M = 1 and D = 26 → 26 January

Why it happens: the three multipliers are 5, 4 and 5, and 5 × 4 × 5 = 100. So by the end the month has been pushed into the hundreds place, while the day D — at most 31 — needs only the tens and units places. The two never mix. Once the constant 165 is taken away, the last two digits of what remains are the day and everything before them is the month.

Did you know? the 165 also comes straight out of the steps: the 6 gets multiplied by 4 and then by 5, giving 6 × 20 = 120, and the 9 gets multiplied by 5, giving 45. And 120 + 45 = 165.

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