# CRE 1 - Strategy Games | DI - Games & Tournaments

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**Answer the next 3 questions based on the information given below:**

Two people Surya and Prithvi are playing a game Some balls are kept on the floor. A person can pick 1, 2 or 3 balls in a turn. Each player takes alternate turns. The one who picks the last ball loses the game. Assume that each player plays intelligently and with an intention to win.

**CRE 1 - Strategy Games | DI - Games & Tournaments**

If Prithvi managed to win the game though Surya began the game, what could have been the minimum number of balls greater than 5 that were lying on the floor?

- (a)
7

- (b)
9

- (c)
13

- (d)
15

Answer: Option B

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**Explanation** :

When two players play a game such that

• they pick balls alternately,

• each player can pick balls from 1 to n,

• last player to pick loses.

The winning strategy in such scenario is to leave ((n+1)k + 1) marbles in your turn.

Whoever picks marbles from ((n+1)k + 1) marbles, he/she will lose.

For Surya to lose the game he should have picked the first time from (4k + 1) marbles.

From the options given, 9 and 13 are of the form 4k + 1. The least of these is 9.

Hence, option (b).

Workspace:

**CRE 1 - Strategy Games | DI - Games & Tournaments**

If Prithvi were to begin the game and win it, what is the number of balls on the floor that is impossible for the game to have begun with?

- (a)
13

- (b)
14

- (c)
15

- (d)
16

Answer: Option A

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**Explanation** :

When two players play a game such that

• they pick balls alternately,

• each player can pick balls from 1 to n,

• last player to pick loses.

The winning strategy in such scenario is to leave (4k + 1) marbles in your turn.

Whoever picks marbles from (4k + 1) marbles, he/she will lose.

For Prithvi to win the game he should not pick marbles from (4k + 1) form.

From the options given, 13 is of the form 4k + 1.

∴ Surya should not pick from 13 marbles.

Hence, option (a).

Workspace:

**CRE 1 - Strategy Games | DI - Games & Tournaments**

If the number of balls on the floor is 47 and Surya is going to start the game then how many balls should he pick in his first turn to ensure his win, irrespective of what Prithvi does in the entire game remaining?

- (a)
3

- (b)
2

- (c)
1

- (d)
Either A or B

Answer: Option B

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**Explanation** :

When two players play a game such that

• they pick balls alternately,

• each player can pick balls from 1 to n,

• last player to pick loses.

The winning strategy in such scenario is to leave (4k + 1) marbles in your turn.

Whoever picks marbles from (4k + 1) marbles, he/she will lose.

For Surya to win the game he should leave (4k + 1) marbles.

Number just less than 47 of (4k + 1) form is 45.

∴ Surya should pick 2 marbles.

Hence, option (b).

Workspace:

**Answer the next 3 questions based on the information given below:**

Two people Ronak and Samyukta are playing a game. Some pebbles are kept on the floor. A person can pick 1, 2 or 3 pebbles in a turn. Each player takes alternate turns. The one who picks the last pebbles wins the game. Assume that each player plays intelligently and with an intention to win.

**CRE 1 - Strategy Games | DI - Games & Tournaments**

What is the winning strategy that a player employs to win the game?

- (a)
leave marbles in the form 4k + 1

- (b)
leave marbles in the form 4k

- (c)
None of these

- (d)
Cannot be determined

Answer: Option B

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**Explanation** :

When two players play a game such that

• they pick balls alternately,

• each player can pick balls from 1 to n,

• last player to pick wins.

The winning strategy in such scenario is to leave 4k marbles in your turn.

Hence, option (c).

Workspace:

**CRE 1 - Strategy Games | DI - Games & Tournaments**

The number of pebbles lying on the floor is between 20 & 23 (both inclusive) and it is Samyukta’s turn to pick. If she picks 2 pebbles to ensure her win, how many pebbles were lying on the floor?

Answer: 22

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**Explanation** :

When two players play a game such that

• they pick balls alternately,

• each player can pick balls from 1 to n,

• last player to pick wins.

The winning strategy in such scenario is to leave 4k marbles in your turn.

If Samyukta picked 2 marbles to leave 4k marbles it means she would’ve picked from 4k + 2 marbles.

Between 20 and 23, only 22 is of the form 4k + 2.

∴ Samyukta picked from 22 marbles.

Hence, 22.

Workspace:

**CRE 1 - Strategy Games | DI - Games & Tournaments**

If there are 61 pebbles lying on the floor and it is Ronak’s turn to start how many pebbles must he pick in order to ensure her win?

Answer: 1

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**Explanation** :

When two players play a game such that

• they pick balls alternately,

• each player can pick balls from 1 to n,

• last player to pick wins.

The winning strategy in such scenario is to leave 4k marbles in your turn.

Whoever picks marbles from 4k marbles, he/she will lose.

For Ronak to win the game he should leave marbles in the form of 4k.

Number just less than 61 of 4k form is 60.

∴ Ronak should pick 1 pebble.

Hence, 1.

Workspace:

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