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NCERT Solutions Ganita Prakash Chapter 3 In-text Questions — Games and Winning Strategies

Book page 71 & 72 Updated on2026-09-05

Q1.
Rules for Game #1: The first player says 1, 2 or 3. Then the two players take turns adding 1, 2, or 3 to the previous number said. The first player to reach 21 wins! Which player can always win if they play correctly? What is the pattern of numbers that the winning player should say?
Answer

The first player can always win. The winning numbers to say are

1, 5, 9, 13, 17, 21

Start by saying 1. After that, whatever your friend adds, you add the amount that makes the total go up by 4 from your last number.

Friend addsYou addTotal goes up by
134
224
314
Why it works: work backwards from 21. To win you must say 21, so you must leave your friend at 17, 18, 19 or 20 — that is, you should say 17. Working back in the same way gives 13, 9, 5 and 1. Whoever says 1 first controls the whole game, and the first player can always say 1.
Try This: the key number is 4 = 3 + 1, that is, the largest allowed addition plus 1. This is the secret of the whole family of such games.
Q2.
Rules for Game #2: The first player says a number between 1 and 10. Then the two players take turns adding a number between 1 and 10 to the previous number said. The first player to reach 99 wins! Which player can always win? What is the pattern of numbers that the winning player should say this time?
Answer

This time the second player always wins. The winning numbers are the multiples of 11:

11, 22, 33, 44, 55, 66, 77, 88, 99

Here the largest allowed addition is 10, so the key number is 10 + 1 = 11. Whatever the first player says (some number x from 1 to 10), the second player replies with 11 − x, bringing the total to 11.

First player says123…910
Second player adds1098…21
Total111111…1111

Repeating this, the second player reaches 22, 33, 44 … and finally 99 — and wins.

Why the first player cannot win: the first player can only say 1 to 10, so he can never say 11. He must always hand over a total that is not a multiple of 11, and the second player instantly restores the pattern.
Q3.
Make your own variations of this game — decide how much one can add at each turn, and what number is the winning number. Then play your game several times, and figure out the winning strategy and which player can always win!
Answer

The rule for every such game is the same. If a player may add anything from 1 to k, then the magic step is k + 1.

Winning numbers = the target, then target − (k + 1), then target − 2(k + 1), and so on.
  • If the first winning number in that backward list can be said on the first move, the first player wins.
  • If the list ends exactly at the target being a multiple of (k + 1), the second player wins.
Gamek + 1Numbers to sayWinner
Add 1–3, reach 2141, 5, 9, 13, 17, 21first player
Add 1–10, reach 991111, 22, …, 99second player
Add 1–3, reach 2044, 8, 12, 16, 20second player
Add 1–5, reach 3161, 7, 13, 19, 25, 31first player
Add 1–4, reach 2555, 10, 15, 20, 25second player
Tip: divide the target by (k + 1). If the remainder is 0, the second player wins; otherwise the first player wins by saying that remainder on the very first move.
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